Are there homomorphisms of group algebras that don't come from a group homomorphism? 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.

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.
