In spring 2020, the chemical education community faced an abrupt transition from in-person to online classes, which also necessitated online assessments. Building upon an existing three-semester study (F17, S19, and F19) using Rasch modeling and classical testing theory to improve in-person multiple choice exams, this study investigates the impact of online exams (F20, F21, and F22) on assessment quality and student performance in an undergraduate General Chemistry II class. The Cronbach's alpha and fraction of very good/good questions were found to dramatically increase across the first two semesters (F17 and S19) and then largely plateaued for subsequent exams, regardless of in-person or online test administration. Through the use of linking questions (i.e., repeated questions from semester to semester) and equating procedures, the results indicated that (1) there was not an obvious or uniform increase or decrease in the exam quality or student performance when switching from in-person to online exams and (2) there was no evidence for an increased prevalence of cheating in the unproctored online exam relative to the prior in-person exams. While this data set is not sufficient to make any universal claims, this case study's outcomes suggest that concerns about increased cheating on unproctored online exams are not inherently founded.