Class group of the ring of invariants of an exponential map on an affine normal domain

Let k be a field and let B be an affine normal domain over k. Let $$\phi $$ be a non-trivial exponential map on B and let $$A = B^{\phi }$$ be the ring of $$\phi $$ -invariants. Since A is factorially closed in B, $$A = K \cap B$$ where K denotes the field of fractions of A. Hence A is a Krull domain. We investigate here a relation between the class group $$\mathrm{Cl}(A)$$ of A and the class group $$\mathrm{Cl}(B)$$ of B. In this direction, we give a sufficient condition for an injective group homomorphism from $$\mathrm{Cl}(A)$$ to $$\mathrm{Cl}(B)$$ . We also give an example to show that $$\mathrm{Cl}(A)$$ may not be realized as a subgroup of $$\mathrm{Cl}(B)$$ .