Let $G$ be a connected reductive group over an algebraically closed field $k$ (with nice enough characteristic), and let $\sigma:G\to G$ be a finite order automorphism of $G$. The connected component $G_0$ of the fixed-point subgroup $G^\sigma$ is also a reductive group. By a theorem of Steinberg, there exists a Borel subgroup $B\subset G$ and maximal torus $T\subset B$ such that both $B$ and $T$ are $\sigma$-stable: ie: $\sigma(B)=B$ and $\sigma(T)=T$. Furthermore, $B_0=G_0\cap B$ is a Borel of $G_0$ and $T$ may be chosen so that $T_0=G_0\cap T$ is a maximal torus of $G_0$.

Now in this set up, $\sigma$ acts on the roots $\Phi=\Phi(G,T)$, preserving the positive and simple roots $\Delta\subset \Phi^+\subset \Phi$ determined by $B$. It also acts on the Weyl group $W=W(G,T)$, and there is a natural subgroup $W_0\subset W$ such that $W_0=W(G_0,T_0)$. This is contained in the fixed-point set $W^\sigma$, but is generally smaller. Finally, let $l$ (resp. $l_0$) be the length function of $W$ (resp. $W_0$).

My question is:

For $w\in W_0\subset W$ what is known about the comparison between the lengths $l_0(w)$ and $l(w)$?

Obviously, $l_0(w)\leq l(w)$, but I am curious about when stronger estimates may hold. I would be interested in any reference where such questions are considered.

A case I am particularly interested in is when $\sigma^2=1$, and there is a maximal torus $A\subset G$ such that $\sigma(a)=a^{-1}$ for all $a\in A$. Such involutions are called split, and in this case it seems like we can choose $(B,T)$ so that $$2l_0(w)\leq l(w),$$

which would have some nice consequences. I've verified this for type $A$, $G_2$, and a couple other small rank cases, but haven't found a general argument. Is this estimate known, or clearly false?