q$ can be conveniently split into three mutually disjoint classes. The first class consists of graphs admitting an imprimitive subgroup of automorphisms with blocks of size $p$ - it coincides with $(q,p)$-metacirculants \cite{AP82a}. The second class consists of graphs admitting an imprimitive subgroup of automorphisms with blocks of size $q$ but no imprimitive subgroup of automorphisms with blocks of size $p$ - it coincides with the class of socalled Fermat graphs, which are certain $q$-fold covers of $K_p$ where $p$ is a Fermat prime \cite{MS2}. The third class consists of vertex-transitive graphs with no imprimitive subgroup of automorphisms. Following \cite[Theorem~2.1]{MS5} the theorem below gives a complete classification of connected vertex-transitive graphs of order $pq$ (see also \cite{PX,PWX}). We would like to remark, however, that there is an additional family of primitive graphs of order $91=7\cdot 13$ that was not covered neither in \cite{MS5} nor in \cite{PX}. This is due to a missing case in Liebeck - Saxl's table \cite{LS85} of primitive group actions of degree $mp$, $m

q$, must be one of the following: \begin{enumerate}[(i)] \itemsep=0pt \item a metacirculant, \item a Fermat graph, \item a generalized orbital graph associated with one of the groups in Table~\ref{tab:groups}. \end{enumerate} \end{theorem} \begin{table}[h!] {\footnotesize \begin{center} \begin{tabular}{|c|c||c|c|c|} \hline & & & & \\ Row & $soc\ G$ & $(p,q)$ & $action$ & $comment$ \\ & & & & \\ \hline\hline 1 & $P\Omega^\epsilon(2d,2)$ & $(2^d-\epsilon,2^{d-1}+\epsilon)$ & singular & $\epsilon=+1:$ $d$ Fermat prime \\ & & & $1$-spaces & $\epsilon=-1:$ $d-1$ Mersenne prime\\ \hline %& & & $1$-spaces & $p$ a Fermat prime \\ \hline 2 & $M_{22}$ & $(11,7)$ & see Atlas & \\ \hline 3 & $A_7$ & $(7,5)$ & triples & \\ \hline 4 & $\PSL(2,61)$ & $(61,31)$ & cosets of & \\ & & & $A_5$ & \\ \hline 5 & $\PSL(2,q^2)$ & $(\frac{q^2+1}{2},q)$ & cosets of & $q \geq 5$ \\ & & & $\PGL(2,q)$ & \\ \hline 6 & $\PSL(2,p)$ & $(p,\frac{p+1}{2})$ & cosets of & $p \equiv1\,(mod\,4)$ \\ & & & $D_{p-1}$ & $p \geq 13$ \\ \hline 7 & $\PSL(2,13)$ & $(13,7)$ & cosets of & missing in \cite{LS85}\\ & & $ $ & $A_4$ & \\ \hline %{\color{red}8} & $\PGL(2,13)$ & $(13,7)$ & {\color{red}cosets of} & \\ % & & & $S_4$ & \\ \hline \end{tabular} \caption{\small \label{tab:groups} Primitive groups of degree $pq$ without imprimitive subgroups and with non-isomorphic genera\-lized orbital graphs.} \end{center}} \end{table} The existence of Hamilton cycles in graphs given in Theorem~\ref{the:main1}(i) and (ii) was proved, respectively, in \cite{AP82a} and \cite{DM92}. It is the aim of this paper to make the next step towards proving the existence of Hamilton cycles in every connected vertex-transitive of order a product of two primes with the exception of the Petersen graph, by showing existence of Hamilton cycles in graphs arising from Row 5 of Table~\ref{tab:groups}. \begin{theorem} \label{the:main} Vertex-transitive graphs arising from the action of $\PSL(2,q^2)$ on $\PGL(2,q)$ given in Row 5 of Table~\ref{tab:groups} are hamiltonian. \end{theorem} %Our aim is therefore to show that graphs %arising from primitive group action given in Row 5 of %Table~\ref{tab:groups} %have a Hamilton cycle. The existence of Hamilton cycles needs to be proved for all connected generalized orbital graphs arising from these actions. Recall that a generalized orbital graph is a union of basic orbital graphs. Since the considered action is primitive and hence the corresponding basic orbital graphs are connected, it suffices to prove the existence of Hamilton cycles solely in basic orbital graphs of this action. This is done in Section~\ref{sec:PGL}. The method used is for the most part based on the socalled lifting cycle technique \cite{A89,KM09,DM83}. Lifts of Hamilton cycles from quotient graphs which themselves have a Hamilton cycle are always possible, for example, when the quotienting is done relative to a semiregular automorphism of prime order and when the corresponding quotient multigraph has two adjacent orbits joined by a double edge contained in a Hamilton cycle. This double edge gives us the possibility to conveniently ``change direction" so as to get a walk in the quotient that lifts to a full cycle above. By \cite[Theorem~3.4]{M81} a vertex-transitive graph of order $pq$, $q