Question about preimage of conjugate class of quotient group

Question

$G$ is a group, $K \triangleleft G,\ |K|=2,\ \overline G=G/K$.

Suppose $\overline C$ is a conjugate class of $\overline G$, $S$ is preimage of $\overline C$ in $G$. Then one of these two holds:

(1) $S=C$ is a single conjugate class with $|C|=2|\overline C|$.

(2) $S=C_1 \cup C_2$ is union of two conjugate classes with $|C_1|=|C_2|=|\overline C|$.

My problem:

What does "$\overline C$ is a conjugate class of $\ \overline G$" mean?

Since conjugate class of $\overline G$ in $\overline G$ is just $\overline G$ itself, I'm not sure what this $\overline C$ refers to.

Answer 1

$\bar{C}$ being a conjugacy class of $\bar{G}$ just means, that $\bar{C}\in\bar{G}$, i.e. $\bar{C}=g\circ K$ for some $g\in G$. The preimage then is: $$\{g\in G: g\circ K=\bar{C}\}$$ Elements of a quotient structure are almost always called conjugacy classes, as they are defined via an equivalence relation :)

Answer 2

According to the hypothesis $\vert K\vert = 2$, $\overline{G}$ is having $n/2$ elements if $n$ is the order of $G$.

Suppose $\overline C$ is a conjugate class of $\overline G$ means that $\overline{C}$ is one of those $n/2$ elements. $\overline G$ is not an element of $\overline G$ by the way, as your last sentence seems to mean.