I'm studying the group action of a group $G$ on a set $X=\{1,\dots,n\}$
For example taking a straightforward example with $G \leq S_4$:
$$X = \{1,2,3,4\} \quad G=\{e_4,(1,2),(3,4),(1,2)(3,4)\}$$ We can construct a group multiplication table based on the action of $G$ in $X$ to determine the group action type.
$$\begin{array}{|l|llll|} \hline & 1 & 2 & 3 & 4 \\ \hline \sigma_1 & 1 & 2 & 3 & 4 \\ \sigma_2 & 2 & 1 & 3 & 4 \\ \sigma_3 & 1 & 2 & 4 & 3 \\ \sigma_4 & 2 & 1 & 4 & 3 \\ \hline \end{array}$$
At first glance is not transitive since $\sigma.x\neq y \quad x,y\in X$ for example $\nexists \sigma\in G$ such that $\sigma \cdot 1 = \sigma(1) = 4$.
It is faithful: given $\sigma,\rho \in G$ there exists $x$ such that $\sigma\cdot x \neq \rho\cdot x$, formally, every row is different by at least one element.
It is not free, so it doesn't satisfy $\sigma \cdot x = \rho \cdot x \iff \sigma = \rho = e$. For example $\sigma_2(1) = \sigma_4(1)$ breaks the rule.
Summarizing, I want a general understanding of group action types. It seems that types of group actions follow a pattern if we observe the table of the action of $G$ on $X$.
Transitive: each column has each $x\in X$ once.
Faithful: each row is different by at least one element.
Free: each row is entirely different to others. (As a consequence every permutation is a derangement since it has no fixed points)
Regular: All the listed above, so the table forms a symmetric matrix.
I can give you a group with regular action, so the table follows the aforementioned criteria:
$$G=\{e_4, (1,2)(3,4),(1,3)(2,4),(1,4)(2,3)\}$$
$$\begin{array}{|l|llll|} \hline & 1 & 2 & 3 & 4 \\ \hline \sigma_1 & 1 & 2 & 3 & 4 \\ \sigma_2 & 2 & 1 & 4 & 3 \\ \sigma_3 & 3 & 4 & 1 & 2 \\ \sigma_4 & 4 & 3 & 2 & 1 \\ \hline \end{array}$$
As you can see it satisfied the criteria above.
Q: What if $X=G$? Would be that an automorphism ($G \times G \to G$)? If it's the case, then do bijections from $G$ to $G$ give the corresponding permutation?
Q:All the groups above were subgroups of the symmetric group on n symbols. If that wasn't the case, how would an example look like?
Q: Can we determine the action type by observing if the properties are satisfied in the table?
Q: Is there any alternative method to identify the group action of a G-set?