This article is concerned with the question whether a subgroup $G$ of a Gromov hyperbolic group $H$ admits proper conjugation, i.e.\ if there exists an element $\alpha\in H$ such that the conjugate $\alpha G\alpha^{-1}$ is a proper subgroup of $G$. In the context of Kleinian groups, i.e.\ discrete isometry groups of hyperbolic $n$-space for $n\ge 2$, examples of proper conjugation groups were given by Jørgensen, Marden and Pommerenke in 1981; Matsuzaki and Yabuki proved in 2009 that groups of divergence type (i.e.\ groups for which the Poincar{é} series diverges at the critical exponent) do not admit a proper conjugation in the group of all isometries of hyperbolic $n$-space. So in particular no geometrically finite Kleinian group admits proper conjugation. \par In the present article the class of groups is enlarged to isometry groups of a proper Gromov hyperbolic space instead of hyperbolic $n$-space. The first main result of the paper is the following: \par Theorem 1.4. If $G$ acts by isometries on a proper Gromov hyperbolic space $X$ with finite critical exponent and a quasiconvex orbit in $X$, then $G$ does not admit a proper conjugation in the group of all isometries of $X$. \par When considering ${\rm CAT}(-1)$-spaces instead of arbitrary Gromov hyperbolic spaces, the conclusion of the above theorem can be obtained with a weaker assumption on $G$ than being quasiconvex cocompact (i.e.\ having a quasiconvex orbit in $X$): \par Theorem 1.5. If $G$ acts uniformly properly discontinuously by isometries on a proper ${\rm CAT}(-1)$-space $X$ and if $G$ is of divergence type, then $G$ does not admit a proper conjugation in the group of all isometries of $X$.