Certain number on the groups of self homotopy equivalences

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 $\text{E}(X)$ be the group of self-homotopy equivalences of $X$. We denote by ${\text{A}}_{\mathrm{♯}}^k(X)$ 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, $[f]\in {\text{A}}_{\mathrm{♯}}^k(X)$ if and only if ${\pi }_i(f):{\pi }_i(X)\to {\pi }_i(X)$ is an isomorphism for $i=0,1,\cdots ,k$. Then, $\text{E}(X)\subseteq {\text{A}}_{\mathrm{♯}}^k(X)\subseteq [X,X]$ for a nonnegative integer $k$. Moreover, for a connected CW-complex $X$, we have $\text{E}(X)={\text{A}}_{\mathrm{♯}}(X)$. In this paper, we study the properties of ${\text{A}}_{\mathrm{♯}}^k(X)$ and discuss the conditions under which $\text{E}(X)={\text{A}}_{\mathrm{♯}}^k(X)$ 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.