Human perceptions of fairness in (semi-)automated decision-making (ADM) constitute a crucial building block toward developing human-centered ADM solutions. However, measuring fairness perceptions is challenging because various context and design characteristics of ADM systems need to be disentangled. Particularly, ADM applications need to use the right degree of automation and granularity of data input to achieve efficiency and public acceptance. We present results from a large-scale vignette experiment that assessed fairness perceptions and the acceptability of ADM systems. The experiment varied context and design dimensions, with an emphasis on who makes the final decision. We show that automated recommendations in combination with a final human decider are perceived as fair as decisions made by a dominant human decider and as fairer than decisions made only by an algorithm. Our results shed light on the context dependence of fairness assessments and show that semi-automation of decision-making processes is often desirable.