  Let $\mc G$ be a reductive group over an algebraically closed field of
characteristic $p>0$. We study homogeneous $\mc G$-spaces that are induced from
the $G\times G$-space $G$, $G$ a suitable reductive group, along a parabolic
subgroup of $\mc G$. We show that, under certain mild assumptions, any (normal)
equivariant embedding of such a homogeneous space is canonically Frobenius
split compatible with certain subvarieties and has an equivariant rational
resolution by a toroidal embedding. In particular, all these embeddings are
Cohen-Macaulay.
  Examples are the $G\times G$-orbits in normal reductive monoids with unit
group $G$. Our class of homogeneous spaces also includes the open orbits of the
well-known determinantal varieties and the varieties of (circular) complexes.
  We also show that all $G$-orbit closures in a spherical variety which is
canonically Frobenius split are normal.
  Finally we study the Gorenstein property for the varieties of circular
complexes and for a related reductive monoid.
