<p>We present progress in the study of a Halpern-type iteration for certain quasi-nonexpansive mappings. We demonstrate that several algorithms designed to solve the split common fixed point problem with multiple output sets can be viewed as special cases of the split common fixed point problem with a single output set. Additionally, we show that some existing strong convergence theorems for both the split common fixed point problem and the split common fixed point problem with multiple output sets can be derived from the results of the first author in 2010. Furthermore, for a specific subclass of quasi-nonexpansive mappings, we relax the assumptions on iteration parameters. We provide examples demonstrating that these improvements yield better numerical outcomes than previous results. Finally, we include a counterexample that highlights the limitations of certain conditions proposed by Jirakitpuwapat et al. in their work on the split common fixed point problem (Jirakitpuwapat et al. in Mathematics 7:226, 2019).</p>