  We study isometric actions of Lie $2$-groups on Riemannian groupoids by
exhibiting some of their immediate properties and implications. Firstly, we
prove an existence result which allows both to obtain 2-equivariant versions of
the Slice Theorem and the Equivariant Tubular Neighborhood Theorem and to
construct bi-invariant groupoid metrics on compact Lie $2$-groups. We provide
natural examples, transfer some classical constructions and explain how this
notion of isometric $2$-action yields a way to develop a 2-equivariant Morse
theory on Lie groupoids. Secondly, we give an infinitesimal description of an
isometric Lie $2$-group action. We define an algebra of transversal
infinitesimal isometries associated to any Riemannian $n$-metric on a Lie
groupoid which in turn gives rise to a notion of geometric Killing vector field
on a quotient Riemannian stack. If our Riemannian stack is separated then we
prove that the algebra formed by such geometric Killing vector fields is always
finite dimensional.
