Let functions $\varphi(g)$ map the elements of a group $G$ to some set ${\cal{L}}$ whose nature is unimportant (say, $\mathbb{C}^N$). The space of all these functions will be denoted with ${\cal{L}}^{G}\,$: $$ {\cal{L}}^{G}~=~\left\{~\varphi~~~{\Large{|}}~~~\varphi\,:~G\,\longrightarrow\,{\cal{L}}\right\} $$ Consider a subgroup $K<G$ and its representation $D(K)$ by linear operators on ${\cal{L}}^{G}\,$: $$ D:\quad K~\longrightarrow~{\rm{GL}}({\cal{L}}^{G})~.\qquad\qquad\qquad\qquad\qquad\qquad\qquad (1) $$ For each fixed $k\in K$, we thus obtain a mapping $$ D_k~:~~~{\cal{L}}^{G} ~\longrightarrow~ {\cal{L}}^{G}~~~,\quad \varphi~\longrightarrow~\varphi^{\,\prime}~~. $$ In application to a group element $g$, this can be written down as $$ [D_k \varphi](g) = \varphi^{\,\prime}(g)~~~ $$ or simply $$ D_k \varphi(g) = \varphi^{\,\prime}(g)~~,\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad (2) $$ while in application to $\,g^{-1}\,$ this reads: $$ D_k \varphi(g^{-1}) = \varphi^{\,\prime}(g^{-1})~~.\qquad\qquad\qquad\qquad\qquad\qquad\quad\quad (3) $$
QUESTION 1.
Without specifying a particular nature of the representation $D(K)$, is it possible to judge on some general grounds if $D^{-1}$ is similar to $D\,$? -- i.e., if there exits a transformation $F$ of the space, rendering $F^{\,-1}\,D_k\,F\,=\,D_{k^{\,-1}}\,$ for $\,\forall k\in K\,$?
QUESTION 2.
We can identify the group $G$ with its manifold, using some parameterisation $$ g~=~g(\alpha)\;\;, $$ so expression (2) becomes some algebraic relation involving the functions $\,\varphi\,$, $\,\varphi^{\,\prime}\,$ and the coordinates $\,\alpha(k)\,$, $\,\alpha(g)\,$.
After we carry out the inversion of all the group elements ($\,g\,\longrightarrow\,g^{\,-1}\,$, $\,k\,\longrightarrow\,k^{\,-1}\,$, etc) will equation (2) turn into $$ D_{k^{\,-1}} \varphi(-g) = \varphi^{\,\prime}(-g)~~?\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad (4) $$ ... and if yes, will that not be in contradiction to (3) ?