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There is a related old paper by Tits on normalisers of tori, but my copy is long gone and I'm not sure whether the splitting issues had been addressed there. In the case of $E_7$, the sequence does not split over $W$ if the argument below is correct.

Indeed suppose that it does. Then $W$ is a subgroup of $G={\rm Aut}(g)$ and for every $\alpha\in \Delta$ we have an involution $\theta_\alpha\in G$ corresponding to the reflection $s_\alpha\in W$. We look at the fixed point algebra $g^{\theta_\alpha}$. If $s_\alpha(\beta)=\beta$ for $\beta\in\Delta$ then ($\theta_\alpha$ being an involution) for any $e_\beta\in g_\beta$ we have that $\theta_\alpha(e_\beta)=c_\alpha(\beta)e_\beta$, where $c_\alpha(\beta)=\pm 1$. Now the centraliser $C_W(s_\alpha)$ contains a reflection subgroup of type $A_1 + D_6$ which acts ttransitively on the set $\Delta_\alpha$ of all roots orthogonal to $\alpha$. From this it is immediate that $c_\alpha(\beta)=c_\alpha(\gamma)$ for all $\beta,\gamma\in\Delta_\alpha$. As we can find $\beta,\gamma\in\Delta_\alpha$ with $\beta+\gamma\in\Delta_\alpha$ and $c_\alpha(\beta+\gamma)=1$, we deduce that $c_\alpha(\beta)=1$ for all $\beta\in\Delta_\alpha$.

Now we can compute $\dim\ g^{\theta_\alpha}$. Each pair of distinct roots $(\beta, s_\alpha(\beta))$ contributes $1$ to $\dim\ g^{\theta_\alpha}$. As $|\Delta_\alpha|=60$ and $|\Delta|=126$, the number of such pairs is $33$ and the above discussion shows that $\dim\ g^{\theta_\alpha}=6+60+33=99$. But it is well known (and goes back to E. Cartan's work on simple Lie algebras over $\mathbb{R}$) that there are three conjugacy classes of involutions in $G$ whose fixed point algebras have dimensions $63$, $69$ and $79$. So the extension does not split over $W$.

There is a related old paper by Tits on normalisers of tori, but my copy is long gone and I'm not sure whether the splitting issues had been addressed there. In the case of $E_7$, the sequence does not split over $W$ if the argument below is correct.

Indeed suppose that it does. Then $W$ is a subgroup of $G={\rm Aut}(g)$ and for every $\alpha\in \Delta$ we have an involution $\theta_\alpha\in G$ corresponding to the reflection $s_\alpha\in W$. We look at the fixed point algebra $g^{\theta_\alpha}$. If $s_\alpha(\beta)=\beta$ for $\beta\in\Delta$ then ($\theta_\alpha$ being an involution) for any $e_\beta\in g_\beta$ we have that $\theta_\alpha(e_\beta)=c_\alpha(\beta)e_\beta$, where $c_\alpha(\beta)=\pm 1$. Now the centraliser $C_W(s_\alpha)$ contains a reflection subgroup of type $A_1 + D_6$ which acts ttransitively on the set $\Delta_\alpha$ of all roots orthogonal to $\alpha$. From this it is immediate that $c_\alpha(\beta)=c_\alpha(\gamma)$ for all $\beta,\gamma\in\Delta_\alpha$. As we can find $\beta,\gamma\in\Delta_\alpha$ with $\beta+\gamma\in\Delta_\alpha$ and $c_\alpha(\beta+\gamma)=1$, we deduce that $c_\alpha(\beta)=1$ for all $\beta\in\Delta_\alpha$.

Now we can compute $\dim\ g^{\theta_\alpha}$. Each pair of distinct roots $(\beta, s_\alpha(\beta))$ contributes $1$ to $\dim\ g^{\theta_\alpha}$. As $|\Delta_\alpha|=60$ and $|\Delta|=126$, the number of such pairs is $33$ and the above discussion shows that $\dim\ g^{\theta_\alpha}=6+60+33=99$. But it is well known (and goes back to E. Cartan's work on simple Lie algebras over $\mathbb{R}$) that there are three conjugacy classes of involutions in $G$ whose fixed point algebras have dimensions $63$, $69$ and $79$. So the extension does not split over $W$.

As for the second part of the question, it seems likely to me that the involution of the extended Dynkin diagram of $\Delta$ can be lifted to an involution of $G$. It has the form $w_0w_1$ where $w_0$ is the longest element of $W$ and $w_1$ is that of the Weyl group of the $E_6$-subdiagram. Let $l$ be the corresponding Levi subalgebra of $g$. One just needs to find an involution in $N_G(l)$ which acts as the nontrivial graph automorphism on $[l,l]$ and as $-1$ on the centre of $l$.

There is a related old paper by Tits on normalisers of tori, but my copy is long gone and I'm not sure whether the splitting issues had been addressed there. In the case of $E_7$, the sequence does not split over $W$ if the argument below is correct.

Indeed suppose that it does. Then $W$ is a subgroup of $G={\rm Aut}(g)$ and for every $\alpha\in \Delta$ we have an involution $\theta_\alpha\in G$ corresponding to the reflection $s_\alpha\in W$. We look at the fixed point algebra $g^{\theta_\alpha}$. If $s_\alpha(\beta)=\beta$ for $\beta\in\Delta$ then ($\theta_\alpha$ being an involution) for any $e_\beta\in g_\beta$ we have that $\theta_\alpha(e_\beta)=c_\alpha(\beta)e_\beta$, where $c_\alpha(\beta)=\pm 1$. Now the centraliser $C_W(s_\alpha)$ is a reflection group of type $A_1 + D_6$ and hence acts ttransitively on the set $\Delta_\alpha$ of all roots orthogonal to $\alpha$. From this it is immediate that $c_\alpha(\beta)=c_\alpha(\gamma)$ for all $\beta,\gamma\in\Delta_\alpha$. As we can find $\beta,\gamma\in\Delta_\alpha$ with $\beta+\gamma\in\Delta_\alpha$ and $c_\alpha(\beta+\gamma)=1$, we deduce that $c_\alpha(\beta)=1$ for all $\beta\in\Delta_\alpha$.

Now we can compute $\dim\ g^{\theta_\alpha}$. Each pair of distinct roots $(\beta, s_\alpha(\beta))$ contributes $1$ to $\dim\ g^{\theta_\alpha}$. As $|\Delta_\alpha|=60$ and $|\Delta|=126$, the number of such pairs is $33$ and the above discussion shows that $\dim\ g^{\theta_\alpha}=6+60+33=99$. But it is well known (and goes back to E. Cartan's work on simple Lie algebras over $\mathbb{R}$) that there are three conjugacy classes of involutions in $G$ whose fixed point algebras have dimensions $63$, $69$ and $79$. So the extension does not split over $W$.

There is a related old paper by Tits on normalisers of tori, but my copy is long gone and I'm not sure whether the splitting issues had been addressed there. In the case of $E_7$, the sequence does not split over $W$ if the argument below is correct.

Indeed suppose that it does. Then $W$ is a subgroup of $G={\rm Aut}(g)$ and for every $\alpha\in \Delta$ we have an involution $\theta_\alpha\in G$ corresponding to the reflection $s_\alpha\in W$. We look at the fixed point algebra $g^{\theta_\alpha}$. If $s_\alpha(\beta)=\beta$ for $\beta\in\Delta$ then ($\theta_\alpha$ being an involution) for any $e_\beta\in g_\beta$ we have that $\theta_\alpha(e_\beta)=c_\alpha(\beta)e_\beta$, where $c_\alpha(\beta)=\pm 1$. Now the centraliser $C_W(s_\alpha)$ contains a reflection subgroup of type $A_1 + D_6$ which acts ttransitively on the set $\Delta_\alpha$ of all roots orthogonal to $\alpha$. From this it is immediate that $c_\alpha(\beta)=c_\alpha(\gamma)$ for all $\beta,\gamma\in\Delta_\alpha$. As we can find $\beta,\gamma\in\Delta_\alpha$ with $\beta+\gamma\in\Delta_\alpha$ and $c_\alpha(\beta+\gamma)=1$, we deduce that $c_\alpha(\beta)=1$ for all $\beta\in\Delta_\alpha$.

Now we can compute $\dim\ g^{\theta_\alpha}$. Each pair of distinct roots $(\beta, s_\alpha(\beta))$ contributes $1$ to $\dim\ g^{\theta_\alpha}$. As $|\Delta_\alpha|=60$ and $|\Delta|=126$, the number of such pairs is $33$ and the above discussion shows that $\dim\ g^{\theta_\alpha}=6+60+33=99$. But it is well known (and goes back to E. Cartan's work on simple Lie algebras over $\mathbb{R}$) that there are three conjugacy classes of involutions in $G$ whose fixed point algebras have dimensions $63$, $69$ and $79$. So the extension does not split over $W$.

There is a related old paper by Tits on normalisers of tori, but my copy is long gone and I'm not sure whether the splitting issues had been addressed there. In the case of $E_7$, the sequence does not split over $W$ if the argument below is correct.

Indeed suppose that it does. Then $W$ is a subgroup of $G={\rm Aut}(g)$ and for every $\alpha\in \Delta$ we have an involution $\theta_\alpha\in G$ corresponding to the reflection $s_\alpha\in W$. We look at the fixed point algebra $g^{\theta_\alpha}$. If $s_\alpha(\beta)=\beta$ for $\beta\in\Delta$ then ($\theta_\alpha$ being an involution) for any $e_\beta\in g_\beta$ we have that $\theta_\alpha(e_\beta)=c_\alpha(\beta)e_\beta$, where $c_\alpha(\beta)=\pm 1$. Now the centraliser $C_W(s_\alpha)$ is a reflection group of type $A_1 + D_6$ and hence acts ttransitively on the set $\Delta_\alpha$ of all roots orthogonal to $\alpha$. From this it is immediate that $c_\alpha(\beta)=c_\alpha(\gamma)$ for all $\beta,\gamma\in\Delta_\alpha$. As we can find $\beta,\gamma\in\Delta_\alpha$ with $\beta+\gamma\in\Delta_\alpha$ and $c_\alpha(\beta+\gamma)=1$, we deduce that $c_\alpha(\beta)=1$ for all $\beta\in\Delta_\alpha$.

Now we can compute $\dim\ g^{\theta_\alpha}$. Each pair of distinct roots $(\beta, s_\alpha(\beta))$ contributes $1$ to $\dim\ g^{\theta_\alpha}$. As $|\Delta_\beta|=60$ and $|\Delta|=126$, the number of such pairs is $33$ and the above discussion shows that $\dim\ g^{\theta_\alpha}=6+60+33=99$. But it is well known (and goes back to E. Cartan's work on simple Lie algebras over $\mathbb{R}$) that there are three conjugacy classes of involutions in $G$ whose fixed point algebras have dimensions $63$, $69$ and $79$. So the extension does not split over $W$.

There is a related old paper by Tits on normalisers of tori, but my copy is long gone and I'm not sure whether the splitting issues had been addressed there. In the case of $E_7$, the sequence does not split over $W$ if the argument below is correct.

Indeed suppose that it does. Then $W$ is a subgroup of $G={\rm Aut}(g)$ and for every $\alpha\in \Delta$ we have an involution $\theta_\alpha\in G$ corresponding to the reflection $s_\alpha\in W$. We look at the fixed point algebra $g^{\theta_\alpha}$. If $s_\alpha(\beta)=\beta$ for $\beta\in\Delta$ then ($\theta_\alpha$ being an involution) for any $e_\beta\in g_\beta$ we have that $\theta_\alpha(e_\beta)=c_\alpha(\beta)e_\beta$, where $c_\alpha(\beta)=\pm 1$. Now the centraliser $C_W(s_\alpha)$ is a reflection group of type $A_1 + D_6$ and hence acts ttransitively on the set $\Delta_\alpha$ of all roots orthogonal to $\alpha$. From this it is immediate that $c_\alpha(\beta)=c_\alpha(\gamma)$ for all $\beta,\gamma\in\Delta_\alpha$. As we can find $\beta,\gamma\in\Delta_\alpha$ with $\beta+\gamma\in\Delta_\alpha$ and $c_\alpha(\beta+\gamma)=1$, we deduce that $c_\alpha(\beta)=1$ for all $\beta\in\Delta_\alpha$.

Now we can compute $\dim\ g^{\theta_\alpha}$. Each pair of distinct roots $(\beta, s_\alpha(\beta))$ contributes $1$ to $\dim\ g^{\theta_\alpha}$. As $|\Delta_\alpha|=60$ and $|\Delta|=126$, the number of such pairs is $33$ and the above discussion shows that $\dim\ g^{\theta_\alpha}=6+60+33=99$. But it is well known (and goes back to E. Cartan's work on simple Lie algebras over $\mathbb{R}$) that there are three conjugacy classes of involutions in $G$ whose fixed point algebras have dimensions $63$, $69$ and $79$. So the extension does not split over $W$.