  Extending the work of Liu--Mubayi--Reiher~\cite{LMR23unif} on hypergraph
Tur\'{a}n problems, we introduce the notion of vertex-extendability for general
extremal problems on hypergraphs and develop an axiomatized framework for
proving strong stability for extremal problems satisfying certain properties.
This framework simplifies the typically complex and tedious process of
obtaining stability and exact results for extremal problems into a much simpler
task of verifying their vertex-extendability.
  We present several applications of this method in generalized Tur\'{a}n
problems including the Erd\H{o}s Pentagon Problem, hypergraph
Tur\'{a}n-goodness, and generalized Tur\'{a}n problems of hypergraphs whose
shadow is complete multipartite. These results significantly strengthen and
extend previous results of Erd\H{o}s~\cite{Erdos62},
Gy\H{o}ri--J\'{a}nos--Simonovits~\cite{GPS91}, Grzesik~\cite{Gre12},
Hatami--Hladk\'{y}--Kr\'{a}\v{l}--Norine--Razborov~\cite{HHKNR13},
Morrison--Nir--Norin--Rz\k{a}\.{z}ewski--Wesolek~\cite{MNNRPW23},
Gerbner--Palmer~\cite{GP22}, and others.
