will this induce extra over head on the Primary database transaction ? Longer statement elapsed time ?
This depends on how pedantic you'd like to be with the question. Technically speaking, since more operations are needed and other areas can be affected, such as ghost cleanup, then the answer would be yes there is extra overhead.
If you're not being pedantic and just ascertaining if generally enabling a secondary to be readable (but no read workload is executed) would the execution time of queries on the primary be about the same then the answer would be, generically, yes.
So it can be both there is extra overhead and there isn't extra overhead depending on how you classify "extra overhead".
-Sean