Research in applied algebra

In the UK, there is the Applied Algebra and Geometry Research Network. You could browse the list of former speakers and abstracts for ideas.

The University of St Andrews has a strong group in Combinatorics and Algebra, with some members (such as Rosemary Bailey) working on computations and applications.

In Ireland, Graham Ellis's group at NUI Galway is very active in the field of computational algebra.

In Leipzig, Germany there is the Max Planck Institute for Mathematics in the Sciences, where in particular Bernd Sturmfels' group works on applications of algebra to non-linear models

A lot of "algebra" is happening in programming language theory and practice nowadays, with knowledge of category theory and type theory really beneficial. Practical applications involve creating certified for correctness programs, and certifying existing programs for correctness. You might have heard about computer-certified proofs of theorems, such as Odd Order Theorem - this essentially falls into the same domain. (Here is the announcement).

For something completely different: symmetries are used in optimisation and machine learning, to reduce dimension etc. And, certainly, there is a lot of algebra in computational (algebraic and "usual") geometry.

Yet another applied topic with a lot of algebra is cryptography and coding theory; among the topics mentioned, it is probably the closest to finite group theory.