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It is proved that among the finite groups of order less than 32 only the tetrahedral group and the binary tetrahedral group are not determined by their endomorphism semigroups in the class of all groups.

We extend the result of P. Puusemp (Idempotents of the endomorphism semigroups of groups. Acta Comment. Univ. Tartuensis, 1975, 366, 76-104) about determinability of finite Abelian groups by their endomorphism semigroups to finite idempotent medial commutative quasigroups.