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We investigate the endomorphism monoids of certain finite $p$-groups of order$p^8$ first studied by Jonah and Konvisser in 1975 as examples for finite$p$-groups with abelian automorphism group, and we show some necessaryconditions for a finite $p$-group to have commutative endomorphism monoid. As aby-product, apart from formulas for the number of conjugacy classes ofendomorphisms of said groups, we will be able to derive the following: Thereexist nonabelian groups with commutative endomorphism monoid, and havingcommutative endomorphism monoid is a group property strictly stronger thanhaving abelian automorphism group. Furthermore, using a result of Curran, thiswill enable us to give, for all primes $p$, examples of finite $p$-groups whichare direct products and have abelian automorphism group.

On the endomorphism monoids of some groups with abelian automorphism group