There is an interesting result of Dade about isomorphism of group algebras:
There exists finite non-isomorphic metabelian groups $G$ and $H$ whose group algebras over any field are isomorphic.
I was looking for the examples of these groups, but I didn't find them; only the result is stated in most of the books or slides online. So, I have one obvious question and one modification of it.
Question 1. What are the (counterexamples) groups constructed by Dade?
Question 2. Is the example by Dade, the smallest order example?
You can find Dade's original paper at http://link.springer.com/article/10.1007/BF01109886.
He gives a construction of pairs of groups of order $p^3q^3$ for arbitrary primes $p,q$ with $q\equiv 1\pmod{p^2}$, so his smallest examples would be of order $1000$ ($p=2$ and $q=5$).
I've no idea whether smaller examples are possible using a different construction.