Let $G=A\ast_C B$ be an amalgamated free product of two infinite subgroups over their intersection.
Is it possible to describe in an 'explicit' way all the subgroups of $G$ of index $2$? Of course, such subgroups are normal.
(1) Recall that subgroups of a free amalgamated product were described by Karrass and Solitar here.
(2) Concerning the subgroup generated by all involutions in the group of automorphisms of the first Weyl algebra (which is the free product of the affine and triangular subgroups), it was shown in Proposition 5b that this group equals the whole group of automorphisms.
(3) This question may be relevant, especially the claim in its answer.
(4) See also this question.