Given a finite group $G$ and a field $k$, we can form the group algebra $kG$ with basis the elements of $G$. There is a natural augmentation $\varepsilon\colon G\to k$ that sends an element to the sum of its coefficients in this basis. This is an algebra map, and is the counit in Hopf algebra terms. It also makes $kG$ into a symmetric frobenius algebra.
It is well known that $kG$ has algebra morphisms (either to itself or another group algebra) that do not come from group homomorphisms. Indeed, non-isomorphic groups can have algebra isomorphisms between their group algebras. But do any of these also preserve the augmentation map?
Specifically, if $f\colon kG\to kH$ is an algebra map, with $G,H$ finite groups, such that $\varepsilon(f(a))=\varepsilon(a)$ for all $a\in kG$, then is $f$ a group homomorphism?
I can't find any that aren't, but have little intuition on how to construct such maps or prove they don't exist.