Let $S_{Gr}$ be the language of groups, $Z$ an arbitrary set that does not contain elements of $\mathcal A_{S_{Gr}}$ (the corresponding alphabet). For each $z \in Z$ take a new constant symbol $c_z$ and let $S_Z := S_{Gr} \cup \{ c_z | z \in Z \}$. Furthermorde consider $\Phi_{Gr}$ as the three axioms for groups and define $V$ as the countable set of variables.
Consider the (henkin) term structure $\mathfrak T_{Z}^{\Phi_{Gr}}$ corresponding to $\Phi_{Gr}$ regarding the language with symbol set $S_Z$ where the equivalence classes for $t \in T^{S_Z}$ are denoted as $\overline t$.
What I want to show is that for each group $\mathfrak G = (G, \times, 1)$ and each map $g : V \cup Z \rightarrow G$ there exists a unique group homomorphism $f : \mathfrak T_{Z}^{\Phi_{Gr}} \rightarrow \mathfrak G$ such that $g(v_i)= f (\overline{v_i}) ~ \forall i \in \mathbb N$ and $g(z) = f(\overline{c_z}) ~\forall z \in Z$
and that if $Z$ is countable $\mathfrak T_{Z}^{\Phi_{Gr}}$ is isomorphic to $ \mathfrak T_{\emptyset}^{\Phi_{Gr}}$ (should be the empty set there) and if there are $Z, Z'$ such that the those term structures are not isomorphic.
I tried to show first that the upper term structure is a group for each $Z$ - isn't that just because $\Phi_{Gr}$ describes the three axioms for groups? For the homomorphism and isomorphic structures I have no idea how to start. Can somebody help?