  Recently there has been much interest in studying the torsion subgroups of
elliptic curves base-extended to infinite extensions of $\mathbb{Q}$. In this
paper, given a finite group $G$, we study what happens with the torsion of an
elliptic curve $E$ over $\mathbb{Q}$ when changing base to the compositum of
all number fields with Galois group $G$. We do this by studying a group
theoretic condition called generalized $G$-type, which is a necessary condition
for a number field with Galois group $H$ to be contained in that compositum. In
general, group theory allows one to reduce the original problem to the question
of finding rational points on finitely many modular curves. To illustrate this
method we completely determine which torsion structures occur for elliptic
curves defined over $\mathbb{Q}$ and base-changed to the compositum of all
fields whose Galois group is $A_4$.
