On the group of self-homotopy equivalences of a 2-connected and 6-dimensional CW-complex

Let $X$ be a \text{\mathrm{2}}-connected and \text{\mathrm{6}}-dimensional CW-complex $X$ such that $H_3(X)\otimes {\mathbb{Z}}_2=0$. This paper aims to describe the group $\text{E}(X)$ of the self-homotopy equivalences of $X$ modulo its normal subgroup ${\text{E}}_{\ast }(X)$ of the elements that induce the identity on the homology groups. Making use of the Whitehead exact sequence of $X$, denoted by WES$(X)$, we define the group $\mathrm{\Gamma }\mathcal{S}(X)$ of $\mathrm{\Gamma }$-automorphisms of WES$(X)$ and we prove that $\text{E}(X)\mathrm{/}{\text{E}}_{\ast }(X)\cong \mathrm{\Gamma }\mathcal{S}(X)\mathrm{.}$