  To a closed wide Lie subgroupoid $\mathbf{A}$ of a Lie groupoid $\mathbf{L}$,
i.e. a Lie groupoid pair, we associate an Atiyah class which we interpret as
the obstruction to the existence of $\mathbf{L}$-invariant fibrewise affine
connections on the homogeneous space $\mathbf{L}/\mathbf{A}$. For Lie groupoid
pairs with vanishing Atiyah class, we show that the left $\mathbf{A}$-action on
the quotient space $\mathbf{L}/\mathbf{A}$ can be linearized.
  In addition to giving an alternative proof of a result of Calaque about the
Poincare-Birkhoff-Witt map for Lie algebroid pairs with vanishing Atiyah class,
this result specializes to a necessary and sufficient condition for the
linearization of dressing actions, and gives a clear interpretation of the
Molino class as an obstruction to the simultaneous linearization of all the
monodromies.
  In the course of the paper, a general theory of connections on Lie groupoid
equivariant principal bundles is developed.
