Suppose that we have a morphism between profinite groups $f: G\_{1}\rightarrow G\_{2}$ such that $f^{\ast}:H\_{cont}^{\ast}(G\_{2},A)\rightarrow H\_{cont}^{\ast}(G\_{1},A) $ is an isomorphism for any finite trivial $G\_{i}$-module $A$. What can we say in general about $f$ ? and in particular when $G\_{i}=\mathrm{Gal}(\overline{k\_{i}}|k\_{i})$, $i=1,2$. 



induced isomorphism in continuous cohomology
Here is one recent result in this direction, taken from I. Efrat and J. Minac, Galois groups and cohomological functors, Trans. of the AMS, <http://www.ams.org/journals/tran/0000-000-00/S0002-9947-2016-06724-0/home.html> :


Let $q=p^s$ be a prime power.
Suppose that $G\_i=\mathrm{Gal}(\overline{K\_i}/K\_i)$ for fields $K\_i$, both containing a root of unity of order $q$, and take the trivial $G\_i$-module $A=\mathbb{Z}/q\mathbb{Z}$ ($i=1,2$).
For a profinite group $G$ consider the closed subgroups $G^{(2)}=G^q[G,G]$ and $G^{(3)}=(G^{(2)})^q[G^{(2)},G]$ (resp., $G^{(3)}=(G^{(2)})^{2q}[G^{(2)},G]$) when $p>2$ (resp., $p=2$). 
Then the map $f^\*$ is an isomorphism if and only if $f$ induces an isomorphism $G\_1/G\_1^{(3)}\cong G\_2/G\_2^{(3)}$ of profinite groups. 


