TITLE: Does such a subgroup exist?
QUESTION [9 upvotes]: I am looking for a certain masa in a $II_1$ factor which is singular and has nontrivial Takesaki invariant.
For this I am looking for an example of an inclusion of groups $H\subset G$ such that:

$G$ is a countable icc (infinite conjugacy class) group
$H$ is abelian
$\forall g\in G-H,\{ hgh^{-1} |h\in H \}$ is infinite
$ |H\backslash G/H| \geq 3$
there exists $g\in G-H$ and $h_1\neq h_2\in H$ with $h_1 g=gh_2$.

Does such an example exist?

REPLY [2 votes]: A more singular example:
Take an infinite index inclusion of abelian groups $K\subset H$. Let a non-trivial group $L$ act on $K$ by automorphisms. Then the amalgamated free product $G=H\underset{K}{\ast} (K\rtimes L)$ satisfies the conditions. Moreover, $L(H)\subset L(G)$ $\,$is a singular masa. On can use the results of Ioana, Peterson and Popa for this, but maybe there are more elementary ways to see this.