Let $\Gamma$ be a Cayley graph over group $K$ and $H$ be a semiregular subgroup of $Aut(\Gamma)$ with two orbits. Then $K=2H$. Is there any other relation between $H$ and $K$ in general? What about special cases?

$\begingroup$ In order to conclude that $K=2H" you should add WITH TWO ORBITS at the end of your first sentence. Now that your question has changed my earlier answer is not applicable and I have deleted it. $\endgroup$ – Nick Gill Nov 11 '12 at 16:17
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In general, you can't say mauch. Take $\Gamma$ to be the complete graph $K_n$. Then $\Gamma$ is a Cayley graph for any group $K$ of order $n$, and any group $H$ of order $n/2$ acts semiregularly on $\Gamma$ with two orbits.

$\begingroup$ Ok. Thanks so much. Do you think may be any relation between this two subgroups in special cases? For example for (noncomplete)Cayley graphs of order twice a prime? $\endgroup$ – majid arezoomand Nov 20 '12 at 13:10