Exercise 7.31 of Jech's Set Theory says:
If $B$ is a Boolean algebra and $A$ is a regular subalgebra of $B$ then the inclusion mapping extends to a (unique) complete embedding of the completion of $A$ into the completion of $B$.
The hint says to use Sikorski's extension theorem, which says the following: Let $A$ be a subalgebra of a Boolean algebra $B$ and let $h$ be a homomorphism from $A$ into a complete Boolean algebra $C$. Then $h$ can be extended to a homomorphism from $B$ into $C$.
I have trouble using the hint to solve the problem: Then $h$ can be extended to a homomorphism from $B$ into $C$. The theorem immediately implies that the inclusion mapping may be extended to a homomorphism of $\bar{A}$ to $\bar{B}$ (their respective completions), but says nothing about the extended map being a complete mapping. I wondered if we may modify Sikorski's extension theorem such that complete homomorphism may be extended a complete homomorphism, but I can't seem to prove it either.