  In this paper we study a notion of HL-extension (HL standing for
Herwig--Lascar) for a structure in a finite relational language $\mathcal{L}$.
We give a description of all finite minimal HL-extensions of a given finite
$\mathcal{L}$-structure. In addition, we study a group-theoretic property
considered by Herwig--Lascar and show that it is closed under taking free
products. We also introduce notions of coherent extensions and ultraextensive
$\mathcal{L}$-structures and show that every countable $\mathcal{L}$-structure
can be extended to a countable ultraextensive structure. Finally, it follows
from our results that the automorphism group of any countable ultraextensive
$\mathcal{L}$-structure has a dense locally finite subgroup.
