  There is a very natural and well-behaved Hopf algebra morphism from
quasisymmetric functions to peak algebra, which we call it Theta map. This
paper focuses on generalizing the peak algebra by constructing generalized
Theta maps for an arbitrary combinatorial Hopf algebra. The image of Theta maps
lies in the odd Hopf subalgebras, so we present a strategy to find odd Hopf
subalgebra of any combinatorial Hopf algebra. We also give a combinatorial
description of a family of Theta maps for Malvenuto-Reutenauer Hopf algebra of
permutations $\operatorname{\mathsf{\mathfrak{S}Sym}}$ whose images are
generalizations of the peak algebra. We also indicate a criterion to check
whether a map is a Theta map. Moreover, precise descriptions of the Theta maps
for the following Hopf algebras will be presented, Hopf subalgebras of
quasisymmetric functions, commutative and co-commutative Hopf algebras, and
theta maps for a Hopf algebra $\mathcal{V}$ on permutations.
