Surjectivity of restriction map between $H^2$

Question

Let $p$ be a prime and $G$ and $H$ be two profinite groups such that $H$ is open distinguished with $G/H$ cyclique of some order $m \equiv 0 \text{ (mod }p \text{)}$. Do we know some conditions on $G$ and $H$ such that the restriction map $$H^2(G, \mathbf{F}_p) \rightarrow H^2(H, \mathbf{F}_p)^{G/H}$$ is surjective ? (with trivial action of $G$ on $\mathbf{F}_p$). Or more generally do we know something "explicit" (meaning computable in the case I'm interested in, e.g. for galois groups with ramification conditions) about its kernel and cokernel ?