Consider the following composite based map $$f: S^2 \xrightarrow{\sim} RP^2/RP^1 \to RP^\infty/RP^1$$ induced by the inclusion of the real projective plane $RP^2$ into infinite real projective space $RP^\infty$.
Consider ordinary homology with coefficients in the cyclic group ${\mathbb{Z}}/2$ of order two. Is the induced map $$(\Omega f)_*:H_q(\Omega S^2; {\mathbb{Z}}/2) \to H_q(\Omega (RP^\infty/RP^1);{\mathbb{Z}}/2)$$ a monomorphism for each $q\ge 0$?