How to determine the invariant of a set?

Question

Consider all colorings of the set $S=\{1,2,3,4,5,7\}$ in $R$, $G$, and $B$. Let $\sigma$ be the permutation given by
$$\sigma= \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 5 & 4 & 7 & 1 & 3 & 6 & 2 \\ \end{pmatrix} $$

Determine $\text{Inv}(\sigma)$

I know what the invariant is, in this case since $1$ goes to $5$, this means that $1$ and $5$ have to be the same color, then $5$ goes to $3$ therefore $5$ has to be the same color as $3$ which also means $3$ and $1$ have to be the same color. Going through this it looks like the invariant is going to consist of the sequences where the $1, 2, 3, 4, 5, \text{ and } 7$ components are going to be the same color and the $6$ component can be any color. The invariant is going to consist of $3^2$ sequences, one example is $(B, B, B, B, B, G, B)$.

Is this correct?

edit:
$Inv (σ)$ is called the invariant set of $σ$ and is the set of all colorings $s$ such that $\sigma(s)=s$. For example, the coloring $(R,G,G,B,B,B,B)$ does not belong to $Inv (σ)$ because $σ(R,G,G,B,B,B,B)=(B,B,B,G,R,B,G)≠(R,G,G,B,B,B,B)$.

Answer 1

Yes, this is fine. It'd be better to phrase the problem like this:

$S_n$ acts on $\{R, G, B\}^n$ by permuting entries. What are the fixed points of $\sigma$, where $\sigma = \ldots$?

In particular I don't think ${\rm Inv}(\sigma)$ is standard notation and I don't think "the invariants" is standard terminology for the fixed point set. Invariants are something else.