  For a connected based space $X$, let $[X,X]$ be the set of all based homotopy
classes of base point preserving self map of $X$ and let $\E(X)$ be the group
of self-homotopy equivalences of $X$. We denote by $\A_{\sharp}^k(X)$ the set
of homotopy classes of self-maps of $X$ that induce an automorphism of
$\pi_i(X)$ for $i=0,1,\cdots,k$. That is, $[f]\in \A_{\sharp}^k(X)$ if and only
if $\pi_i(f):\pi_i(X)\to\pi_i(X)$ is an isomorphism for $i=0,1,\cdots ,k$.
Then, $\E(X)\subseteq\A_{\sharp}^k(X)\subseteq [X,X]$ for a nonnegative integer
$k$. Moreover, for a connected CW-complex $X$, we have $\E(X)=\A_{\sharp}(X)$.
In this paper, we study the properties of $\A_{\sharp}^k(X)$ and discuss the
conditions under which $\E(X)=\A_{\sharp}^k(X)$ and the minimum value of such
$k$. Furthermore, we determine the value of $k$ for various spaces, including
spheres, products of spaces, and Moore spaces.
