  Let $\Gamma$ be a discrete finitely generated group. Let $\hat{M}\to T$ be a
$\Gamma$-equivariant fibration, with fibers diffeomorphic to a fixed even
dimensional manifold with boundary $Z$. We assume that $\Gamma\to \hat{M}\to
\hat{M}/\Gamma$ is a Galois covering of a compact manifold with boundary. Let
$(D^+ (\theta))_{\theta\in T}$ be a $\Gamma$-equivariant family of Dirac-type
operators. Under the assumption that the boundary family is $L^2$-invertible,
we define an index class in the K-theory of the cross-product algebra, $K_0
(C^0 (T)\rtimes_r \Gamma)$. If, in addition, $\Gamma$ is of polynomial growth,
we define higher indeces by pairing the index class with suitable cyclic
cocycles. Our main result is then a formula for these higher indeces: the
structure of the formula is as in the seminal work of Atiyah, Patodi and
Singer, with an interior geometric contribution and a boundary contribution in
the form of a higher eta invariant associated to the boundary family. Under
similar assumptions we extend our theorem to any $G$-proper manifold, with $G$
an \'etale groupoid. We employ this generalization in order to establish a
higher Atiyah-Patodi-Singer index formula on certain foliations with boundary.
Fundamental to our work is a suitable generalization of Melrose
$b$-pseudodifferential calculus as well as the superconnection proof of the
index theorem on $G$-proper manifolds recently given by Gorokhovsky and Lott.
