Changing variable in topological groups

Question

Let $G \subset GL_n(F)$ be a locally compact group endowed with its Haar measure. Two typical automorphism of $G$, involutions even, are the transposition and inversion. Is it clear that we have the right to change variables $$\int_G f(g) dg = \int_G f({}^tg)dg = \int_G f(g^{-1})dg \quad ?$$

Answer 1

The groups $G$ for which you always have$$\int_Gf(g)\,\mathrm dg=\int_Gf\left(g^{-1}\right)\,\mathrm dg,$$and some locally compact groups are not unimodular. One such group is the group $G$ of those matrices of the form$$\begin{bmatrix}x&y\\0&x^{-1}\end{bmatrix},$$with $x\in(0,\infty)$ and $y\in\Bbb R$.

Note that, in general, if $M\in G$, $M^t\notin G$. So, your other question makes no sense.