  Let $(V,W)$ be an exceptional spetsial irreducible reflection group $W$ on a
complex vector space $V$, that is a group $G_n$ for $n \in \{4, 6, 8, 14, 23,
24, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37\}$ in the Shephard-Todd
notation. We describe how to determine some data associated to the
corresponding (split) "spets", given complete knowledge of the same data for
all proper subspetses (the method is thus inductive). The data determined here
is the set Uch$(\mathbb G)$ of "unipotent characters" of $\mathbb G$ and the
associated set of Frobenius eigenvalues, and its repartition into families. The
determination of the Fourier matrices linking unipotent characters and
"unipotent character sheaves" will be given in another paper. The approach
works for all split reflection cosets for primitive irreducible reflection
groups. The result is that all the above data exist and are unique (note that
the cuspidal unipotent degrees are only determined up to sign).
