Structure of the monoid of maps up to homotopy

Question

For a topological space $X$, what do we know about the monoid $[X,X]$ : continuous maps up to homotopy ? For example, what is the structure of $[RP^2,RP^2]$ ? $RP^2$ is the projective space of dimension $2$. I guessed it might be the same as for the universal cover $S^2$ so $\mathbb{Z}$ or a semi-direct product with $\pi_1(RP^2)=\mathbb{Z}/2\mathbb{Z}$.