Monoids of endomorphisms of nonisomorphic groups

For any prime $p$, the endomorphism monoid of $\mathbb{Z}[\frac{1}{p}]$ is a commutative monoid with zero whose submonoid of nonzero elements is the direct sum of a cyclic group of order two (generated by multiplication by $-1$), an infinite cyclic group (generated by multiplication by $p$), and a free commutative monoid on countably many generators (multiplication by other primes).

But different primes give nonisomorphic groups.

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).