Let $\mathbf L=(C,F,P)$ be a first order language.
Let $\mathbf I$ be an interpretation of $\mathbf L$, Let $\xi(\mathbf I)$ denote the structure established by $\mathbf I$, that is $\xi(\mathbf I)=(U_\mathbf I,C_\mathbf I, F_\mathbf I,P_\mathbf I)$, where $U_\mathbf I$ is the universe of discourse of $\mathbf I$.
Is every property of $\xi(\mathbf I)$ expressible by a formula $\gamma\in FORM(\mathbf {L})$ ?
As an example:
Let $\mathbf L=(C=\emptyset,F=\{f^2,g^2\},P=\{p^2\})$.
Let I be an interpretation of $L$ such that $C_I=\emptyset,F=\{f^2(x,y)=x+y,g^2(x,y)=x-y\},P=\{p^2=\{(x,y)\in U_I:x=y\}\}, U_I=\mathbf Z$.
Is there a formula to describe that $(\mathbf Z, f^2,g^2)$ forms a group structure?
Certainly not every property can be described by a formula. For instance, for any cardinal number $\kappa$, there is a property "$U_\mathbf{I}$ has cardinality $\kappa$". There is a proper class of different cardinal numbers, so this gives a proper class of different properties. But there is only a set of possible formulas, or even possible sets of formulas. If your language is countable, there are even only countably many formulas, so for any uncountable collection of properties, most of them cannot be expressed by a formula.
However, in your example, the property that the structure is a group can easily be expressed by a formula, at least if you have equality as a logical symbol. For instance, the existence of an identity element can be expressed as $$\exists u \forall x (x=f(u,x)\wedge x=f(x,u)).$$ The other axioms of a group can be expressed similarly; to get a single formula, you then take the conjunction of the formulas for each axiom.