  A finite \emph{$k$-net} of order $n$ is an incidence structure consisting of
$k\ge 3$ pairwise disjoint classes of lines, each of size $n$, such that every
point incident with two lines from distinct classes is incident with exactly
one line from each of the $k$ classes. Deleting a line class from a $k$-net,
with $k\ge 4$, gives a \emph{derived} ($k-1$)-net of the same order.
  Finite $k$-nets embedded in a projective plane $PG(2,K)$ coordinatized by a
field $K$ of characteristic $0$ only exist for $k=3,4$, see \cite{knp_k}. In
this paper, we investigate $3$-nets embedded in $PG(2,K)$ whose line classes
are in perspective position with an axis $r$, that is, every point on the line
$r$ incident with a line of the net is incident with exactly one line from each
class. The problem of determining all such $3$-nets remains open whereas we
obtain a complete classification for those coordinatizable by a group. As a
corollary, the (unique) $4$-net of order $3$ embedded in $PG(2,K)$ turns out to
be the only $4$-net embedded in $PG(2,K)$ with a derived $3$-net which can be
coordinatized by a group. Our results hold true in positive characteristic
under the hypothesis that the order of the $k$-net considered is smaller than
the characteristic of $K$.
