I'm writing a pdf where I want to show that two groups $G$ and $H$ are isomorphic. To give some context, $G$ is the automorphism group of a commutative semi-ring $M$ and $H$ is the absolute Galois group of the rationals. I can prove that $G$ and $H$ embed in each other, and that $Inn(G)\cong Inn(H)$, the latter being isomorphic to $Aut(H)$ and to $H$ itself.
Can I conclude that $G\cong H$?