  In this paper we provide a local construction of a Sasakian manifold given a
K\"ahler manifold. Obatined in this way manifold we call Sasakian lift of
K\"ahler base. Almost contact metric structure is determined by the operation
of the lift of vector fields - idea similar to lifts in Ehresmann connections.
We show that Sasakian lift inherits geometry very close to its K\"ahler base.
In some sense geometry of the lift is in analogy with geometry of hypersurface
in K\"ahler manifold. There are obtained structure equations between
corresponding Levi-Civita connections, curvatures and Ricci tensors of the lift
and its base. We study lifts of symmetries different kind: of complex
structure, of K\"hler metric, and K\"ahler structure automorphisms. In
connection with $\eta$-Ricci solitons we introduce more general class of
manifolds called twisted $\eta$-Ricci solitons. As we show class of
$\alpha$-Sasakian twisted $\eta$-Ricci solitons is invariant under naturally
defined group of structure deformations. As corollary it is proved that orbit
of Sasakian lift of steady or shrinking Ricci-K\"ahler soliton contains
$\alpha$-Sasakian Ricci soliton. In case of expanding Ricci-K\"ahler soliton
existence of $\alpha$-Sasakina Ricci solition is assured provided expansion
coefficient is small enough.
