Isomorphism problem for the center of modular group algebras

Question

Let $p$ be a prime number, $G,H$ a finite p-Group and $K$ a finite field with $\operatorname{char}(K)=p$. It is well-known that the group $1+\operatorname{rad}(KG)$ is a p-group containing $G$.

My question is: If $Z(G)$ and $Z(H)$ are isomorphic, then $Z(1+\operatorname{rad}(KG))$ and $Z(1+\operatorname{rad}(KH))$ are ismomorphic and vice versa?

Answer 1

see both comments; both implications are wrong.