  In this work, we study various properties of embedded hypersurfaces in
$1+1+2$ decomposed spacetimes with a preferred spatial direction, denoted
$e^{\mu}$, which are orthogonal to the fluid flow velocity of the spacetime and
admit a proper conformal transformation. To ensure a non-vanishing positivity
scalar curvature of the induced metric, we impose that the scalar curvature of
the conformal metric is non-negative and that the associated conformal factor
satisfies $\hat{\varphi}^2+2\hat{\hat{\varphi}}>0$, where $\hat{\ast}$ denotes
derivative along $e^{\mu}$. Firstly, it is demonstrated that such hypersurface
is either Einstein or the twist vanishes on it, and that the scalar curvature
of the induced metric is constant. It is then proved that if the hypersurface
is compact and of Einstein, and admits a proper conformal transformation, then
the hypersurface must be isomorphic to the $3$-sphere, where we make use of
well known results on Riemannian manifolds admitting conformal transformations.
If the hypersurface is not Einstein and has nowhere vanishing sheet expansion,
we show that this conclusion fails. However, with the additional conditions
that the scalar curvatures of the induced metric and the conformal metric
coincide, the associated conformal factor is strictly negative and the third
and higher order derivatives of the conformal factor vanish, the conclusion
follows. Furthermore, additional results are obtained under the conditions that
the scalar curvature of a metric conformal to the induced metric is also
constant. Finally, we consider some of our results in context of locally
rotationally symmetric spacetimes and show that, if the hypersurfaces are
compact and not of Einstein type, then under specified conditions the
hypersurface is isomorphic to the $3$-sphere, where we constructed explicit
examples of proper conformal Killing vector fields along $e^{\mu}$.
