  In this work we extend our study on a link between automaticity and certain
algebraic power series over finite fields. Our starting point is a family of
sequences in a finite field of characteristic $2$, recently introduced by the
first author in connection with algebraic continued fractions. By including it
in a large family of recurrent sequences in an arbitrary finite field, we prove
its automaticity. Then we give a criterion on automatic sequences, generalizing
a previous result and this allows us to present new families of automatic
sequences in an arbitrary finite field.
