For a pointed space X, the subgroups of self-homotopy equivalences $Aut_{sharp}_N(X)$, $Aut_{Omega}(X)$, $Aut_*(X)$ and $Aut_{Sigma}(X)$ are considered, where $Aut_{sharp}_N(X)$ is the group of all self-homotopy classes f of X such that $f_{sharp}=id;:;{pi_i}(X){ ightarrow}{pi_i}(X)$ for all $i{leq}N{leq}{infty}$, $Aut_{Omega}(X)$ is the group of all the above f such that ${Omega}f=id;;Aut_*(X)$ is the group of all self-homotopy classes g of X such that $g_*=id;:;H_i(X){ ightarrow}H_i(X)$ for all $i{leq}{infty}$, $Aut_{Sigma}(X)$ is the group of all the above g such that ${Sigma}g=id$. We will prove that $Aut_{Omega}(X_1{ imes}cdots{ imes}X_n)$ has two factorizations similar to those of $Aut_{sharp}_N(X_1{ imes}cdots{ imes};X_n)$ in reference [10], and that $Aut_{Sigma}(X_1{vee}cdots{vee}X_n)$, $Aut_*(X_1{vee}cdots{vee}X_n)$ also have factorizations being dual to the former two cases respectively.