Meaning of the term $X/H$ and orbits

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I am trying to find representations of the group $G=GL_2(F(t)/t^2) = (M_2(F_p) , + ) \rtimes GL_n(F)$ So I was trying to do exactly what Serre has explained in this section.

I am not quite able to understand what does the following sentence mean ' systems of representativesfor the orbits of $H$ in $X$ '

ALso here what is denoted by $X/H$ ? does it correspond to quotient group ? If it is a group action then what is the action ?

I am confused.

Thanks

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