Let $G$ be an infinite group, and for simplicity, we will assume that $G$ is also countable. Now, with $G$ in mind, we construct a new language $L_G=\{f_{a_i\_},f_{\_a_i}:a_i\in G\}$ where $f_{a_i\_}(x)=a_ix$ and $f_{\_a_i}(x)=xa_i$ (Note that if $G$ is abelian, we can can restrict the language to $\{f_{\_a_i}:a_i \in G\}$) . Let $T=Th_{L_G}(G)$ (i.e. $T$ is the theory of G in the language $L_G$). It is clear that since $G$ is countable, $L$ is also countable. Therefore, $T$ is a countable, complete theory in a countable language. We say that $M$ is 'group-like' if $M\models T$ for some countably infinite group $G$.
Question 1: If $G$ is countably categorical in $L=\{1,\cdot\}$, then is $T$ countably categorical in $L_G$? (My suspicion is no).
Question 2: What do uncountable 'groups-like' models look like?
Question 3: Can anyone give me an example (countable or uncountable) of a model $M$ such that $M \models T$ but $M$ is "radically not a group" (I know that "radically not a group" is not well defined, but I think the intuition is clear).
So, by Levon Haykazyan's first comment, question 1 is answered in the negative.
From this paper, on page 14, we can answer questions 2 and 3.
Example 1.38 states, "Let $G$ be an infinite group of cardinality $\kappa$ say. Let $L = \{\lambda_g : g \in G\}$ where the $\lambda_g$ are unary function symbols. Let $T$ be the theory of free $G$-sets. Then $T$ is categorical in every cardinality $> \kappa$, and complete".
I think this is the fullest answer this question can possibly receive.