The averages of the molecular weight distribution (MWD) and the degree of branching (DB) are calculated for hyperbranched copolymers obtained in the self-condensing vinyl copolymerization (SCVCP) of AB* initiator-monomers ("inimers") with a monomer M in the presence of an f-functional initiator, G*(f). Two cases are considered: (a) batch polymerization when all components are mixed simultaneously; (b) semibatch polymerization, i.e., slow addition of the comonomer mixture to the system. The presence of initiator leads to a considerable narrowing of the MWDs, the nonuniformity of the distribution (U = (M) over bar (w)/(M) over bar (n) - 1) decreasing with initiator functionality, f, similar to normal self-condensing vinyl polymerization (SCVP): U proportional to f(-2) in the batch mode and U = f(-1) in the semibatch mode. In the semibatch process, the presence of comonomer does not further decrease the polydispersity, as compared to the homopolymerization of AB* inimers in the presence of polyinitiator. The degree of branching is only slightly affected by the presence of polyinitiator and is mostly governed by the comonomer content. In contrast to semibatch SCVP with initiator, there is no increase in DB. The calculations are also applied to polymerizations from surface-bound initiators.