The following is a small collection of examples. I am not sure why you are interested in the restriction map being surjective, but I know a big use of restriction maps deals with their kernel, and in fact the intersection of its kernels to all subgroups (called *Essential Cohomology*).

But first, for example by Frobenius reciprocity, the corestriction map $\text{cor}^G_H$ is a homomorphism of $\widehat{H}^\ast(G,A)$-modules. This is better than having the trivial actions of the groups themselves on their cohomology.

Also, if $H$ is a normal subgroup of $G$ and $A$ is a $G$-module, then the conjugation action of $G$ on $(H,A)$ induces an action of $G/H$ on $H^\ast(H,A)$. But yes, $G$ acts trivially on $H^*(G,A)$. I refer you to Ken Brown's *Cohomology of Groups* for more information.

1) By Mislin's theorem, the restriction map from $G$ to $H$ (a subgroup) on mod-p cohomology is an isomorphism if and only if $H$ controls p-fusion in $G$.

2) A rather trivial example: The inclusion $G\hookrightarrow G\times\mathbb{Z}_p$ induces a surjective restriction map (mod-p coefficients) by functoriality.

3) You can also go the other route and write down all the vanishing and nilpotence results on $H^\ast(H,A)$; there is a ton.

[[Edit]]: Now the response to the restriction map not being surjective is much heavier, but you can easily just look some of this stuff up in Ken Brown's textbook, so I don't think it should be asked on this forum.

1) If $G$ is finite and $H$ is an abelian sylow-p subgroup, and $A$ is a trivial $G$-module, then the image of the restriction map is $H^\ast(H,A)^{N_G(H)}$.

2) If $|G:H|<\infty$ and $A$ is a field of characteristic $p$ relatively prime to $|G:H|$, then the restriction map is injective, because $\text{cor}\circ\text{res}(z)=|G:H|z$.

3) If $|G:H|<\infty$ and is invertible in $A$ (a $G$-module), then the restriction map sends $H^\ast(G,A)$ isomorphically onto $H^\ast(H,A)^G$.

4) Let $G$ be a $2$-group. It turns out that $\text{Ker}(\text{res}^G_H)$ is the principal ideal $(x)$, where $|G:H|=2$ and $x\in H^1(G,\mathbb{Z}_2)$ is a homomorphism $x:G\rightarrow \mathbb{Z}_2$ such that $\text{Ker}x=H$. Since the maximal subgroups of a $p$-group are the subgroups of index $p$, we see that every nontrivial element $x$ corresponds to some maximal subgroup $M\subset G$ (which has index $2$) with $\text{Ker}x=M$.

Note, for injective restriction maps, couple it to Mislin's theorem to see when it is not surjective.

You can also take a look at this short paper I wrote of explicit calculations of restriction maps and their kernels: http://arxiv.org/abs/1006.4836.

** In fact, given a collection $\mathcal{H}$ of subgroups of $G$, the ring $H^\ast(G,A)$ is said to be *detected* by $\mathcal{H}$ if all the restriction maps associated to $\mathcal{H}$ are injective. As an example, $H^\ast(G)$ is detected on the set of Sylow subgroups.

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