TITLE: Monoids of endomorphisms of nonisomorphic groups
QUESTION [17 upvotes]: Can monoids of endomorphisms of nonisomorphic groups be isomorphic ?

REPLY [15 votes]: It is proved in $[$1$]$ that the tetrahedral group $A_4$ of order $12$ and the binary tetrahedral group of order $24$ have isomorphic endomorphism monoids.  So this gives a finite example.  It is also the smallest order example.  Moreover, no other finite groups have isomorphic endomorphism monoid to the endomorphism monoid of these two groups without being isomorphic to one of them.
$[$1$]$ P. Puusemp, Groups of order 24 and their endomorphism semigroups, J. Math. Sci. (2007) 144(2) 3980–3992
(link at Springerlink).