  We introduce the $\mathcal{T}$-construction, an endofunctor on the category
of generalized operads as a general mechanism by which various notions of
plethystic substitution arise from more ordinary notions of substitution. In
the special case of one-object unary operads, i.e. monoids, we recover the
$T$-construction of Giraudo. We realize several kinds of plethysm as
convolution products arising from the homotopy cardinality of the incidence
bialgebra of the bar construction of various operads obtained from the
$\mathcal{T}$-construction. The bar constructions are simplicial groupoids, and
in the special case of the terminal reduced operad $\mathsf{Sym}$, we recover
the simplicial groupoid of arXiv:1804.09462, a combinatorial model for ordinary
plethysm in the sense of P\'olya, given in the spirit of Waldhausen $S$ and
Quillen $Q$ constructions. In some of the cases of the
$\mathcal{T}$-construction, an analogous interpretation is possible.
