  We investigate the rank gain of elliptic curves, and more generally, Jacobian
varieties, over non-Galois extensions whose Galois closure has Galois group
permutation-isomorphic to a prescribed group $G$ (in short, "$G$-extensions").
In particular, for alternating groups and (an infinite family of) projective
linear groups $G$, we show that most elliptic curves over (e.g.) $\mathbb{Q}$
gain rank over infinitely many $G$-extensions, conditional only on the parity
conjecture. More generally, we provide a theoretical criterion which allows to
deduce that "many" elliptic curves gain rank over infinitely many
$G$-extensions, conditional on the parity conjecture and on the existence of
geometric Galois realizations with group $G$ and certain local properties.
