Let $G_1$ and $G_2$ be two isomorphic finite groups. Suppose that $\sigma_i:G_i\to G_i$ and $\sigma_i':G_i\to G_i$ be two isomorphisms for $i=1,2$.
When is there an isomorphism $\phi:G_1\to G_2$ such that $\phi\sigma_1=\sigma_2\phi$ and $\phi\sigma_1'=\sigma_2'\phi$?
Thanks for your helps in advance
P.S.
The origin of the problem:
Let $G=G_1\times G_2$ and suppose that $Aut(G)$ denote the automorphism group of $G$. Clearly every $\sigma\in Aut(G)$ such that $\sigma(G_1)=G_1$ and $\sigma(G_2)=G_2$ can induce a group automorphism of $G_1$ and $G_2$.
I try find a bijection $\phi$ from $G_1$ to $G_2$ such that $$\phi(\sigma(g_1))=\sigma(\phi(g_1))$$ for a subgroup $H$ of $Aut(G)$ such that every $\sigma\in H$ has the above property.
If for every $g_1\in G_1$ we have
$$ \phi(\sigma(\phi^{-1}(g_2)))=\sigma(g_2),$$ then it is a bijection with the above property.