Is there some proposition saying how to extend an isomorphism of $k$-vector spaces where $k$ is a field of characteristic $p$ to an isomorphismus of $k[H]$-modules where $H$ is a group of order prime to $p$ ? (This should be some semi-simple case)
My problem arose in the context of Lazard's famous theorem [1]: For a uniformly powerful pro-$p$-group $G$ there is an isomorphism of $\mathbb{F}_p$ vector spaces.
$$H^i(G,\mathbb{F}_p) \longrightarrow \bigwedge\nolimits^i((G/G^p[G,G])^*)$$
Which I like to extend to the case of both sides being $H$-modules.
Thank you! :)
[1] M. Lazard, Groupes analytiques p-adiques, IHES publ. math. 26 (1965)