  In this paper, we prove the Uplifting Theorem. It states the following. Let
$p$ be a prime. Let $G$ be a $(1,1)$-smooth profinite group. Let $A$ be a
perfect $(\mathbb F_p,G)$-algebra. Then, complete flags of semi-linear
representations of $G$ over $A$ lift, to complete flags of semi-linear
representations of $G$ over $\mathbf W_2(A)$. As a corollary, we get the
unconditional liftability of mod p Galois representations (of all fields, of
all dimensions) to mod p^2 Galois representations. The Uplifting Theorem is a
key ingredient, in the new proof of the Norm Residue Isomorphism Theorem,
offered in the next paper of this series.
