  We prove the existence of a potentially diagonalizable lift of a given
automorphic mod $p$ Galois representation $\overline{\rho}:{\rm
Gal}(\overline{F}/F)\longrightarrow {\rm GSp}_4(\overline{\mathbb{F}}_p)$ for
any totally real field $F$ and any rational prime $p>2$ under the adequacy
condition by using automorphic lifting techniques developed by Barnet-Lamb,
Gee, Geraghty, and Taylor. As an application, when $p$ is split completely in
$F$, we prove a variant of Serre's weight conjecture for $\overline{\rho}$. The
formulation of our Serre conjecture is done by following Toby Gee's philosophy.
Applying these results to the case when $F=\mathbb{Q}$ with a detailed study of
potentially diagonalizable, crystalline lifts with some prescribed properties,
we also define classical (naive) Serre's weights. This weight would be the
minimal weight among possible classical weights in some sense which occur in
candidates of holomorphic Siegel Hecke eigen cusp forms of degree 2 with levels
prime to $p$. The main task is to construct a potentially ordinary automorphic
lift for $\overline{\rho}$ by assuming only the adequacy condition. The main
theorems in this paper also extend many results obtained by Barnet-Lamb, Gee
and Geraghty for potentially ordinary lifts and Gee and Geraghty for companion
forms.
