  We establish endpoint estimates for a class of oscillating spectral
multipliers on Lie groups of Heisenberg type. The analysis follows an earlier
argument due to the second and fourth author but requires the detailed analysis
of the wave equation on these groups due to M\"uller and Seeger. We highlight
and develop the connection between sharp bounds for oscillating multipliers and
the problem of determining the minimal amount of smoothness required for
Mihlin-H\"ormander multipliers, a problem that was solved for groups of
Heisenberg type but remains open for other groups.
