  Let $M$ and $N$ be topological spaces, let $G$ be a group, and let $\tau
\colon\thinspace G \times M \to M$ be a proper free action of $G$. In this
paper, we define a Borsuk-Ulam-type property for homotopy classes of maps from
$M$ to $N$ with respect to the pair $(G,\tau)$ that generalises the classical
antipodal Borsuk-Ulam theorem of maps from the $n$-sphere $\mathbb{S}^n$ to
$\mathbb{R}^n$. In the cases where $M$ is a finite pathwise-connected
CW-complex, $G$ is a finite, non-trivial Abelian group, $\tau$ is a proper free
cellular action, and $N$ is either $\mathbb{R}^2$ or a compact surface without
boundary different of $\mathbb{S}^2$ and $\mathbb{RP}^2$, we give an algebraic
criterion involving braid groups to decide whether a free homotopy class $\beta
\in [M,N]$ has the Borsuk-Ulam property. As an application of this criterion,
we consider the case where $M$ is a compact surface without boundary equipped
with a free action $\tau$ of the finite cyclic group $\mathbb{Z}_n$. In terms
of the orientability of the orbit space $M_\tau$ of $M$ by the action $\tau$,
the value of $n$ modulo $4$ and a certain algebraic condition involving the
first homology group of $M_\tau$, we are able to determine if the single
homotopy class of maps from $M$ to $\mathbb{R}^2$ possesses the Borsuk-Ulam
property with respect to $(\mathbb{Z}_n,\tau)$. Finally, we give some examples
of surfaces on which the symmetric group acts, and for these cases, we obtain
some partial results regarding the Borsuk-Ulam property for maps whose target
is $\mathbb{R}^2$.
