Take a closed hemisphere and identify the antipodal points on the equator ,we get $\mathbb RP^{2}$ and inside $ \mathbb RP^{2}$ we have copy of $ \mathbb RP^{1}$.So, what will be the induced map on fundamental group induced by inclusion?
I think ,if we know which homotopy class of loop in $\mathbb RP^{2}$ is generator,then some conclusion can be said easily.