Given a finite group $G$, one can define the group algebra $\mathbb{C}[G]$ as the algebra having the elements of $G$ as a basis, with the multiplication of $G$. Clearly, any group homomorphism induces an algebra homomorphism on the group algebras.
I'm wondering whether one can prove that any algebra homomorphism of two group algebras must always come from a homomorphism of groups.
On
It is well known that the dihedral group of order $8$ and the quaternions have isomorphic group algebras. Taking an isomorphism and restricting it to the basis of the dihedral group given by the elements of the group does not give you the basis of the quaternions given by the elements of the group because if it did this would imply that the dihedral group and the quaternions were isomorphic as groups. So in this sense it does not "come from a group homomorphism".
Let $G = \{e, \sigma\}$ be the group of order $2$. Then $\{e, \sigma\}$ is a basis for $\mathbb C[G]$. The linear map given by $e \mapsto e$ and $\sigma \mapsto -\sigma$ is an automorphism of $\mathbb C[G]$ that does not come from a group homomorphism.