This paper aims to investigate homomorphisms which preserve p-primitive languages. A characterization of p-primitivity-preserving homomorphisms can be detected within finite steps. Also the set of square-freeness-preserving homomorphisms is shown to be a proper subfamily of the set of p-primitivity-preserving homomorphisms. For homomorphisms over an alphabet X with |X|=2, it is also shown that the set of p-primitivity-preserving homomorphisms is a proper subfamily of the set of primitivity-preserving homomorphisms. But it is conjectured to also hold for homomorphisms over an alphabet with more than two letters.
