  Let $\Gamma=\Gamma(N)$ be a principal congruence subgroup of $SL_2(\mathbb
Z)$. In this paper, we extend the theory of modular Hecke algebras due to
Connes and Moscovici to define the algebra $\mathcal Q(\Gamma)$ of quasimodular
Hecke operators of level $\Gamma$. Then, $\mathcal Q(\Gamma)$ carries an action
of "the Hopf algebra $\mathcal H_1$ of codimension $1$ foliations" that also
acts on the modular Hecke algebra $\mathcal A(\Gamma)$ of Connes and Moscovici.
However, in the case of quasimodular forms, we have several new operators
acting on the quasimodular Hecke algebra $\mathcal Q(\Gamma)$. Further, for
each $\sigma\in SL_2(\mathbb Z)$, we introduce the collection $\mathcal
Q_\sigma(\Gamma)$ of quasimodular Hecke operators of level $\Gamma$ twisted by
$\sigma$. Then, $\mathcal Q_\sigma(\Gamma)$ is a right $\mathcal
Q(\Gamma)$-module and is endowed with a pairing $(\_\_,\_\_):\mathcal
Q_\sigma(\Gamma)\otimes \mathcal Q_\sigma(\Gamma)\longrightarrow \mathcal
Q_\sigma(\Gamma)$. We show that there is a "Hopf action" of a certain Hopf
algebra $\mathfrak{h}_1$ on the pairing on $\mathcal Q_\sigma(\Gamma)$.
Finally, for any $\sigma\in SL_2(\mathbb Z)$, we consider operators acting
between the levels of the graded module $\mathbb
Q_\sigma(\Gamma)=\underset{m\in \mathbb Z}{\oplus}\mathcal
Q_{\sigma(m)}(\Gamma)$, where $\sigma(m)=\begin{pmatrix} 1 & m \\ 0 & 1 \\
\end{pmatrix}\cdot \sigma$ for any $m\in \mathbb Z$. The pairing on $\mathcal
Q_\sigma(\Gamma)$ can be extended to a graded pairing on $\mathbb
Q_\sigma(\Gamma)$ and we show that there is a Hopf action of a larger Hopf
algebra $\mathfrak{h}_{\mathbb Z}\supseteq \mathfrak{h}_1$ on the pairing on
$\mathbb Q_\sigma(\Gamma)$.
