  The spectrum of a group is the set of orders of its elements. Finite groups
with the same spectra as the direct squares of the finite simple groups with
abelian Sylow 2-subgroups are considered. It is proved that the direct square
$J_1\times J_1$ of the sporadic Janko group $J_1$ and the direct squares
${^2}G_2(q)\times{^2}G_2(q)$ of the simple small Ree groups ${^2}G_2(q)$ are
uniquely characterized by their spectra in the class of finite groups, while
for the direct square $PSL_2(q)\times PSL_2(q)$ of a 2-dimensional simple
linear group $PSL_2(q)$, there are always infinitely many groups (even solvable
groups) with the same spectra.
