Due to the first (and very helpful) answer I received, I've reformulated the question a little: $G$ and $H$ are now assumed to be $p$-groups.

Let $\{p}$ be a prime, and let $\mathbb{F}_p$ be the field of $\{p}$ elements. Let $G,H$ be finite $p$-groups, and let $\mathbb{k}[G]$ denote the group algebra.

Does $\mathbb{K}[G]\simeq\mathbb{K}[H]$ for some field $\mathbb{K}$ of characteristic $p$, imply $\mathbb{F}_p[G]\simeq\mathbb{F}_p[H]$?