  A group is said to be self-similar provided it admits a faithful state-closed
representation on some regular $m$-tree and the group is said to be transitive
self-similar provided additionally it induces transitive action on the first
level of the tree. A standard approach for constructing a transitive
self-similar representation of a group has been by way of a single virtual
endomorphism of \ the group in question. Recently, it was shown that this
approach when applied to the restricted wreath product $% \mathbb{Z}\wr
\mathbb{Z}$ could not produce a faithful transitive self-similar
representations for any $m\geq 2$ (see, \cite{DS}). In this work we study
state-closed representations without assuming the transitivity condition. This
general action is translated into a set of virtual endomorphisms corresponding
to the different orbits of the action on the first level of the tree. In this
manner, we produce faithful self-similar representations, some of which are
also finite-state, for a number of groups such as $\mathbb{Z}^{\omega}$,
$\mathbb{Z}\wr \mathbb{Z}$ and $(\mathbb{Z} \wr \mathbb{Z}) \wr C_{2}$.
