Given a complex simple Lie algebra $ \mathfrak g $ of type $E_7$, Cartan subalgebra $ \mathfrak h $ and simple roots $\alpha_1,…\alpha_n $, suppose $\pi $ is an involution of the extended Dynkin diagram. I would like to know whether $\pi $ must be induced from an involution of $ \mathfrak g $. Writing $\Delta $ for the root system and W for the Weyl group, since $ Aut(\Delta)/W $ is isomorphic to the group of automorphisms of the Dynkin diagram and this is trivial for $ E_7 $, the answer is "yes" if the map $ Aut(\mathfrak g,\mathfrak h)\rightarrow Aut(\Delta) $ splits over the Weyl group.

That is, it would suffice if there is a subgroup of the automorphism group of $Aut(\mathfrak g, \mathfrak h)$ which is isomorphic to the Weyl group under this mapping.

Is this true? Or do you otherwise know whether every involution of the extended Dynkin diagram for $ E_7 $ must arise from an involution of $ \mathfrak g $?

Thanks very much!