Fraïssé introduced the notion of a $k$-set-homogeneous relational structure. In the present paper the following classes of monounary algebras are described: $\mathcal Sh_2(S)$, $\mathcal Sh_2(S^c)$, $\mathcal Sh_2(P^c)$ —the class of all algebras which are 2-set-homogeneous with respect to subalgebras, connected subalgebras, connected partial subalgebras, respectively, and $\mathcal H_2(S)$, $\mathcal H_2(S^c)$, $\mathcal H_2(P^c)$ —the class of all algebras which are 2-homogeneous with respect to subalgebras, connected subalgebras, connected partial subalgebras, respectively.