  We discuss a fine tuning of the co- and contra-variant transforms through
construction of specific fiducial and reconstructing vectors. The technique is
illustrated on three different forms of induced representations of the
Heisenberg group. The covariant transform provides intertwining operators
between pairs of representations. In particular, we obtain the Zak transform as
an induced covariant transform intertwining the Schr\"odinger representation on
$\mathsf{L}_2(\mathbb{R})$ and the lattice (nilmanifold) representation on
$\mathsf{L}_2\big(\mathbb{T}^2\big)$. Induced covariant transforms in other
pairs are Fock-Segal-Bargmann and theta transforms. Furthermore, we describe
peelings which map the group-theoretical induced representations to convenient
representation spaces of analytic functions. Finally, we provide a condition
which can be imposed on the reconstructing vector in order to obtain an
intertwining operator from the induced contravariant transform.
