Subgroups of $\mathrm{PSL}(2,q)$ of order $2q$ Let $q\equiv 1\pmod 4$. Is it true that $\mathrm{PSL}(2,q)$ has a unique class of conjugate subgroups of order $2q$? 
I looked at the references that appear in this MO question, the only relevant refernce there is Oliver King's notes where he cites the classification given in Dickson's book. The classification theorem there has $22$ items (items (a) to (v)), and it appears that the only item where a subgroup of order $2q$ can appear is item (m), but the description of that item is very mysterious:

(m) a number of classes of conjugate groups of order $q_0d$ for each divisor $q_0$ of $q$ and for certain $d$ depending on $q_0$, all lying inside a group of order $q(q − 1)/2$ for $q$ odd and $q(q − 1)$ for $q$ even;

Using GAP I checked all appropriate $q$'s up to $100$ and in these cases there is a unique conjugacy class of subgroups of order $2q$, isomorphic to $D_{2q}$ when $q$ is prime; and isomorphic to $(C_p)^e\rtimes C_2$ when $q=p^e$. Is this a general fact?  If so, is there a nice interpretation of this subgroup, such as the stabilizer in $\mathrm{PSL}(2,q)$ of something?

Yes, it is true. Let $B$ denote the normalizer of a Sylow $p$-subgroup of $G = {\rm PSL}(2,q)$, where $q = p^{a}$ for the odd prime $p.$ Then $B$ may be taken as the image (mod $\pm I$ of the group of upper triangular matrices of determinant $1.$ Let $M$ be a subgroup of $G$ of order $2q$. Then $M$ has a normal $2$-complement, which is a Sylow $p$-subgroup of $G.$ Since we are only concerned with $M$ up to conjugacy, we my suppose that $U$ is a common Sylow $p$-subgroup of $B$ and $M.$ Note that this places $M$ inside $B$ as $U \lhd M.$ Now a complement to $U$ in $B$ is the image of the diagonal matrices of determinant $1,$ which is cyclic. Now $B/U$, being cyclic, has a unique subgroup of order $2$. Hence $B$ has a unique subgroup of order $2|U|$ by the isomorphism theorems. Hence this must be $M.$ In other words, in general, every subgroup of $G$ of order $2|U|$ is $G$-conjugate to the unique subgroup of $B$ of order $2|U|$. This is the image (mod scalars) of the group of upper triangular matrices of determinant $1$ with (not necessarily) primitive $4$-th roots of unity on the diagonal.
