Let $G\_{1}$ and $G\_{2}$ be two groups. Suppose that we have a morphism $\mathbb{Z}[G\_{1}]\rightarrow \mathbb{Z}[G\_{2}] $ of bialgebras is it true that this morphism comes from a morphism of groups $G\_{1}\rightarrow G\_{2}$ ?
In case when the answer is "no", is it true that if $\mathbb{Z}[G\_{1}]\rightarrow \mathbb{Z}[G\_{2}] $ is an isomorphism of rings then there exists an isomorphism of bialgebras $\mathbb{Z}[G\_{1}]\rightarrow \mathbb{Z}[G\_{2}] $ ?



groupring morphisms and bialgebra
I will show that any bialgebra homomorphism $\mathbb QM\_1\to \mathbb QM\_2$ of monoid algebras is induced by a monoid homomorphism $M\_1\to M\_2$. This will imply what the OP wants.


An element $g$ of a bialgebra is called group-like if $\Delta(g)=g\otimes g$ and $\eta(g)=1$ where $\eta$ is the counit. It is well known that the group-like elements are linearly independent and form a monoid (cf. Lemma 2.1 <http://www.math.wisc.edu/~passman/balgebra.pdf>). 


If $M$ is a monoid then the elements of $M$ are group-like in the monoid algebra and from the above linear independence they are the only group-like elements. 


Since bialgebra morphisms preserve grouplikes it follows any bialgebra morphism of monoid algebras is inducted by a monoid homomorphism. Hence any bialgebra morphism of group algebras is induced by a group homomorphism. 


