  Let $M$ be a topological space that admits a free involution $\tau$, and let
$N$ be a topological space. A homotopy class $\beta \in [ M,N ]$ is said to
have the Borsuk-Ulam property with respect to $\tau$ if for every
representative map $f: M \to N$ of $\beta$, there exists a point $x \in M$ such
that $f(\tau(x))= f(x)$. In this paper, we determine the homotopy class of maps
from the $2$-torus $T^2$ to the Klein bottle $K^2$ that possess the Borsuk-Ulam
property with respect to any free involution of $T^2$ for which the orbit space
is $K^2$. Our results are given in terms of a certain family of homomorphisms
involving the fundamental groups of $T^2$ and $K^2$. This completes the
analysis of the Borsuk-Ulam problem for the case $M=T^2$ and $N=K^2$, and for
any free involution $\tau$ of $T^2$.
