  Let $(\pi,X)$ be a depth-$0$ admissible smooth complex representation of a
$p$-adic reductive group that splits over an unramified extension. In this
paper we develop the theory necessary to study the wavefront set of $X$ over a
maximal unramified field extension of the base $p$-adic field. In the final
section we then apply these methods to compute the geometric wavefront set of
spherical Arthur representations of split $p$-adic reductive groups. In this
case we see how the wavefront set over a maximal unramified extension can be
computed using perverse sheaves on the Langlands dual group.
