  In this article, we discuss necessary condition of conditional dilation for
both completely non-unitary (c.n.u) $\Gamma_{n}$-contractions and c.n.u
$\mathbb E$-contractions. Consider two tuples, $(A_1, \dots, A_{n-1})$ and
$(B_1, \dots, B_{n-1})$, of operators defined on two Hilbert spaces. One of the
principal goals is to identify a necessary and a sufficient condition
guaranteeing the existence of a c.n.u $\Gamma_n$-contraction $(S_1, \dots,
S_{n-1},P)$ such that $(A_1, \dots, A_{n-1})$ becomes the $\mathcal{F}_O$-tuple
of $(S_1, \dots, S_{n-1},P)$ and $(B_1, \dots,B_{n-1})$ becomes the
$\mathcal{F}_O$-tuple of $(S_1^*,\dots, S_{n-1}^*,P^*)$. Also for given two
pairs of operators $(F_1,F_2)$ and $(G_1,G_2)$ defined on two Hilbert spaces,
we examine when there is a c.n.u $\mathbb E$-contraction $(A,B,P)$ such that
$(F_1,F_2)$ becomes the $\mathcal{F}_O$-pair of $(A,B,P)$ and $(G_1,G_2)$
becomes the $\mathcal{F}_O$-pair of $(A^*,B^*,P^*)$.
