Certain homotopy properties related to $\text{map}({\mathrm{\Sigma }}^n\mathbb{C}{P}^2,S^m)$

For given spaces $X$ and $Y$, let $map(X,Y)$ and $ma{p}_{\ast }(X,Y)$ be the unbased and based mapping spaces from $X$ to $Y$, equipped with compact-open topology respectively. Then let $map(X,Y;f)$ and $ma{p}_{\ast }(X,Y;g)$ be the path component of $map(X,Y)$ containing $f$ and $ma{p}_{\ast }(X,Y)$ containing $g$, respectively. In this paper, we compute cohomotopy groups of suspended complex plane ${\pi }^{n+m}({\mathrm{\Sigma }}^n\mathbb{C}{P}^2)$ for $m=6,7$. Using these results, we classify path components of the spaces $map({\mathrm{\Sigma }}^n\mathbb{C}{P}^2,S^m)$ up to homotopy equivalent. We also determine the generalized Gottlieb groups $G_n(\mathbb{C}{P}^2,S^m)$. Finally, we compute homotopy groups of mapping spaces $map({\mathrm{\Sigma }}^n\mathbb{C}{P}^2,S^m;f)$ for all generators $[f]$ of $[{\mathrm{\Sigma }}^n\mathbb{C}{P}^2,S^m]$, and Gottlieb groups of mapping components containing constant map $map({\mathrm{\Sigma }}^n\mathbb{C}{P}^2,S^m;0)$.