  In this work, we shall provide an $F(R)$ gravity theoretical framework for
solving the $H_0$-tension. Specifically, by exploiting the $F(R)$ gravity
correspondence with a scalar-tensor theory, we shall provide a condition in
which when it is satisfied, the $H_0$-tension is alleviated. The condition that
remedies the $H_0$-tension restricts the corresponding $F(R)$ gravity, and we
present in brief the theoretical features of the constrained $F(R)$ gravity
theory in both the Jordan and Einstein frames. The condition that may remedy
the $H_0$-tension is based on the existence of a metastable de Sitter point
that occurs for redshifts near the recombination. This metastable de Sitter
vacuum restricts the functional form of the $F(R)$ gravity in the Jordan frame.
We also show that by appropriately choosing the $F(R)$ gravity, along with the
theoretical solution offered for the $H_0$-tension problem, one may also
provide a unified description of the inflationary era with the late-time
accelerating era, in terms of two extra de Sitter vacua. We propose a new
approach to $F(R)$ gravity by introducing a new class of integral $F(R)$
gravity functions, which may be wider than the usual class expressed in terms
of elementary $F(R)$ gravity functions. Finally, the Einstein frame
inflationary dynamics formalism is briefly discussed.
