= Solution
The <Fano plane> has as points the seven nonzero <vectors> of $\mathbb F_2^3$, because every one-dimensional <vector subspace> has just one nonzero <vector>. Its lines are the seven two-dimensional <vector subspaces>. Every line has three points, every point lies on three lines, two points determine the line $\{u,v,u+v\}$, and two lines meet in one point.
The two order-24 classes are therefore <stabilizer subgroups> of points and of lines. Each is isomorphic to $S_4$: fixing a <vector> gives the block group $\mathbb F_2^2\rtimes GL_2(2)$, the full affine group on four points; inverse transpose gives the dual line <stabilizer subgroup>. Their distinct fixed-point behavior in part (i) distinguishes the two <conjugacy classes>.
For a point $p$ and line $L$, $G_p\cap G_L$ stabilizes the ordered pair $(p,L)$. There are $7\cdot3=21$ incident pairs and $7\cdot4=28$ nonincident pairs. A change of <basis> takes any pair to a fixed model of its type, so each type is a single orbit. <Orbit-stabilizer theorem> gives the <Fano point-line stabilizer intersection>:
$$
\boxed{|G_p\cap G_L|=
\begin{cases}
8,&p\in L,\\
6,&p\notin L.
\end{cases}}
$$
Both occur: take $L=\langle e_1,e_2\rangle$ and choose $p=\langle e_1\rangle$ or $p=\langle e_3\rangle$.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2010/iii/paper-4-fano-plane.png]
{title=Fano plane with incident and nonincident point-line pairs}
For the extension, identify <vectors> and covectors by the standard pairing. The <inverse-transpose automorphism> exchanges a point with its orthogonal plane and exchanges the two order-24 classes. Thus it is outer. The <Fano plane duality group> $\overline G=G\rtimes\langle\tau\rangle$ has order 336 and acts on points and lines, preserving incidence while allowing their types to be swapped.
The <stabilizer subgroup> of an unordered point-line pair has twice the corresponding ordered-pair <stabilizer subgroup>: some element outside $G$ swaps the pair, since duality produces a pair of the same incidence type and $G$ is transitive on such pairs. Hence these <stabilizer subgroups> have orders 16 and 12. Also every outside conjugate of a Sylow 7-subgroup can be moved back by an element of $G$, giving
$$
|N_{\overline G}(P)|=2|N_G(P)|=42.
$$
We prove these exhaust the <maximal subgroups> other than $G$. Let $K$ be such a <maximal subgroup> and $H=K\cap G$. Since $K$ is not contained in $G$, $H$ is normal in $K$ and $|K:H|=2$. If $7\mid|H|$, part (i) gives a unique Sylow 7-subgroup $P$ of $H$. It is characteristic in $H$, hence normalized by $K$. Therefore maximality forces $K=N_{\overline G}(P)$.
Suppose $7\nmid|H|$ and $H\ne1$. The classification argument in part (i) says $H$ fixes a point or a line. Because $H$ is normal in $K$, an element outside $G$ swaps its fixed-point and fixed-line sets, so it fixes at least one of each. Write
$$
d=\dim V^H,\qquad d^*=\dim(V^*)^H.
$$
Fixed lines are kernels of fixed nonzero covectors, since the only nonzero scalar is one. Duality gives $2^d-1=2^{d^*}-1$, hence $d=d^*\in\{1,2\}$.
If $d=1$, there is a unique fixed point and unique fixed line; $K$ preserves their unordered pair. If $d=2$, put $W=V^H$ and $A=((V^*)^H)^\perp$. For every $h\in H$, $\ker(h-I)$ contains the plane $W$ and $\operatorname{im}(h-I)$ lies in the line $A$. Over $\mathbb F_2$ there is only one nonzero <linear map> $V/W\to A$, so $H$ has order two. Invertibility forces $A\subseteq W$: for $h=I+v f$ one needs $f(v)=0$. Thus $A$ and $W$ are its unique image point and kernel line. These are canonically swapped by duality, so $K$ again preserves their unordered incident pair.
If $H=1$, then $K$ has order two, generated by an involutory duality $u$. For any point $p$, the pair $\{p,u(p)\}$ is preserved by $u$. Hence $K$ lies in an order-12 or order-16 pair <stabilizer subgroup> and is not maximal. We have exhausted all cases.
The pair <stabilizer subgroups> are themselves maximal: a proper maximal overgroup is one of the just-listed types, and divisibility excludes orders 12 inside 16 or 42, and 16 inside 12 or 42. The order-42 <normalizer> is maximal by the unique-Sylow argument. Incidence types give one <conjugacy class> each; Sylow conjugacy gives one <normalizer> class. Thus, apart from the normal index-two <subgroup> $G$,
$$
\boxed{\overline G:\text{ three maximal-subgroup classes, of orders }12,\ 16,\ 42.}
$$
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