  We consider the effect of performing an elementary equivalence as defined by
Okuyama on a group block of form $F(C_p \times C_p)\rtimes C_r$, for a field
$F$ of characteristic $p$. If $I=\{0,1,2,\dots r-1\}$ is the set of residues
corresponding to the simple modules of $FC_r$, the elementary equivalence is
determined by a proper, non-empty subset $I_0 \subset I$, and the corresponding
elementary tilting complex is completely determined by $I_0$.
  We give a catalog of homogeneous maps between irreducible components of the
elementary tilting complex. When the subset $I_0$ is an interval, we prove that
the maps in the catalog are sufficient to describe all homogeneous maps between
two irreducible components.
