Denote $X=\mathbb{R}P^2$ to be the 2-dimensional real projective space, see definition here wikipedia.
Also from the above link, we know that it has a CW complex structure with one 0-cell $x_0$, one 1-cell and one 2-cell.
Then suppose we have a group homomorphism $\phi: \pi_1(X,x_0)\rightarrow \pi_1(Y,y_0)$, where $Y$ is a path-connected space, prove that we can find a map $f:(X,x_0)\rightarrow (Y,y_0)$, such that $f_*=\phi.$
Remark: this is a special case of one homework, I know that we can first define a map $f:S^1\rightarrow Y$ to be $f(a)=b,$ where $\phi([a])=[b]$ and $S^1$ is the 1-cell, $[a]$ is the generator of $\pi_1(X)=\mathbb{Z}/{2\mathbb{Z}}$, but how to extend it to the whole $X$, i.e., to the whole 2-cell?