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The spectrum of a group is the set of orders of its elements. Finite groupswith the same spectra as the direct squares of the finite simple groups withabelian 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 ${^2}G_2(q)$ areuniquely characterized by their spectra in the class of finite groups, whilefor the direct square $PSL_2(q)\times PSL_2(q)$ of a 2-dimensional simplelinear group $PSL_2(q)$, there are always infinitely many groups (even solvablegroups) with the same spectra.

On recognition of the direct squares of the simple groups with abelian Sylow 2-subgroups