  We consider the homotopy types of $PD_4$-complexes $X$ with fundamental group
$\pi$ such that $c.d.\pi=2$ and $\pi$ has one end. Let $\beta=\beta_2(\pi;F_2)$
and $w=w_1(X)$. Our main result is that (modulo two technical conditions on
$(\pi,w)$) there are at most $2^\beta$ orbits of $k$-invariants determining
"strongly minimal" complexes (i.e., those with homotopy intersection pairing
$\lambda_X$ trivial). The homotopy type of a $PD_4$-complex $X$ with $\pi$ a
$PD_2$-group is determined by $\pi$, $w$, $\lambda_X$ and the $v_2$-type of
$X$. Our result also implies that Fox's 2-knot with metabelian group is
determined up to TOP isotopy and reflection by its group.
