$G$-CW complex structure of universal a $\mathcal{F}$-space Let $G$ be a finite group and $H$ be an abelian subgroup of $G$. Let $\mathcal{F}$ be a family of all subgroups of $H$ , i.e. $\mathcal{F}= \{K : K \leq H\}$ Define universal $\mathcal{F}$-space $E\mathcal{F}$ as in Peter May's Alaska Notes.
Question 1: What is $G$-CW complex structure of $E\mathcal{F}$ ? 
Question 2: Is the $n$-th skeleton of $E\mathcal{F}$ is $G$-homotopy equivalence with $G/H \ast G/H \ast \cdots \ast G/H$ (n+1 topological join)?
Any hint or reference will be appreciated.
Thank you.
 A: Firstly, your family $\mathcal{F}$ is not closed under conjugation if $H$ is not normal. Depending on what you want to do, this may not be an issue.
There are two references for the construction of $E\mathcal{F}$ that I know of (but I would be glad to hear of more). The first is tom Dieck's book Transformation groups, where in Chapter I.6 he states that a model for $E\mathcal{F}$ is given by the infinite join 
$$
E\mathcal{F} = X \ast X \ast \cdots
$$
where $X=\bigsqcup G/H_a$ is a disjoint union of orbit types. This is a $G$-CW complex in an obvious way, and suggests that the answer to your second question is no.
The second reference is Lück's Transformation groups and algebraic $K$-theory, available from the author's webpage (scroll down to Books). Apparently, using the results in Chapter 2 of that book, one can build a $G$-CW complex model for $E\mathcal{F}$ by an iterative process of attaching $G$-cells. The construction is not carried out explicitly, however, and I'm not sure of the details (perhaps this is the topic for a separate MO question).
