  Let $H \subseteq G$ be an inclusion of $p$-adic Lie groups. When $H$ is
normal or even subnormal in $G$, the Hochschild-Serre spectral sequence implies
that any continuous $G$-module whose $H$-cohomology vanishes in all degrees
also has vanishing $G$-cohomology. With an eye towards applications in $p$-adic
Hodge theory, we extend this to some cases where $H$ is not subnormal, assuming
that the $G$-action is analytic in the sense of Lazard.
