  Let $\mathcal{E}(X)$ be the group of homotopy classes of self homotopy
equivalences for a connected CW complex $X$. We observe two classes of maps
$\mathcal{E}$-maps and co-$\mathcal{E}$-maps. They are defined as the maps
$X\to Y$ that induce the homomorphisms $\mathcal{E}(X)\to \mathcal{E}( Y)$ and
$\mathcal{E}(Y)\to \mathcal{E}(X)$, respectively. We give some rationalized
examples related to spheres, Lie groups and homogeneous spaces by using
Sullivan models. Furthermore, we introduce an $\mathcal{E}$-equivalence
relation between rationalized spaces $X_{\mathbb{Q}}$ and $Y_{\mathbb{Q}}$ as a
geometric realization of an isomorphism $\mathcal{E}(X_{\mathbb{Q}})\cong
\mathcal{E}(Y_{\mathbb{Q}})$.
