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The number of elements in ${\mathcal O}_{gH}$ is the index of $H\cap gHg^{-1}$ in $H$. Your condition 1 means that there is an element $g$ such that $|H:H\cap gHg^{-1}|\lt |H:H\cap g^{-1}Hg|=|gHg^{-1}: H\cap gHg^{-1}|$. So you want $H\cap gHg^{-1}$ to have different but finite indices in $H$ and in $gHg^{-1}$. The best way to achieve what you want (including 2 and 3) is to consider a group $H$ with two isomorphic subgroups $U,V$ of different finite indices, then consider the HNN extension $\langle H,g\mid gUg^{-1}=V\rangle$. For example, take $H=F_2\times F_3$, $U=U_1\times U_2$ where $U_1$ is of index 10 in $F_2$, $U_2$ is of index 12 in $F_3$ (in that case, by Schreier's formula, the rank of $U_1$ is 11, the rank of $U_2$ is 25), $V=V_1\times V_2$ with $V_1$ of index 24, $V_2$ of index $5$ (then the rank of $V_1$ is 25 and the rank of $V_2$ is 11). In that case $U$ is isomorphic to $V$. The other properties should follow from the general properties of HNN extensions.