  We present new theoretical results on the existence of intersective
polynomials with certain prescribed Galois groups, namely the projective and
affine linear groups $PGL_2(\ell)$ and $AGL_2(\ell)$ as well as the affine
symplectic groups $AGSp_4(\ell):=(\mathbb{F}_\ell)^4 \rtimes GSp_4(\ell)$. For
further families of affine groups, existence results are proven conditional on
the existence on certain tamely ramified Galois extensions. We also compute
explicit families of intersective polynomials for certain non-solvable groups.
