Problem 20.58 from the Kourovka notebook, particularly the first part, reads
Is it true that for every positive integer $n$ there is a recognizable group that is the $n$-th direct power of a non-abelian simple group? (by the way: "recognizability" of a group is defined as follows: Let $\omega(G)$ denote the set of element orders of a finite group $G$. A finite group $G$ is said to be recognizable (by spectrum) if every finite group $H$ with $\omega(H) = \omega(G)$ is isomorphic to $G$).
If I am interpreting the problem correctly, a clear (and very natural) proof strategy would proceed by showing that for every $n$, there is a recognizable group $G$ such that there exists an isomorphism from $G$ to the $n$-th direct power of some non-abelian simple group $H$, which would, of course then be $QED$ (the details of the proof itself are non-trivial, I have no idea how such a proof would work out, it seems rather ambitious).
My confusion is with the wording of the problem statement. The sentence "there is a recognizable group that is the $n$-th direct power of a non-abelian simple group" makes me think that showing that there exists such a group that is merely isomorphic to the $n$-th direct power of a non-abelian simple group, for any $n$ isn't in fact sufficient. Am I missing something?
If the statement is true with "isomorphic to" then it is also true with "is", so this doesn't really matter.
Also note that if a group is isomorphic to a direct product of two simple groups then it is equal to the internal direct product of two simple subgroups, so it is doubtful whether here is really any difference in meaning between the two formulations.