I want to do systematic review and meta-analysis but I am facing difficulties: how to take adjusted odds ratio (AOR) when studies classify the explanatory variables differently? For example one study may put the AOR for education in three ways — "No education" as a reference category, "Primary education" with AOR (CI) , Secondary education with AOR(CI) — and the other study might put e.g. "Primary education and below" as a reference, "Secondary and above" with AOR (CI). So in the first study there are two adjusted odds ratios and in the second one only one adjusted ratio. Is it possible to take the crude odds ratio by calculating manually for meta-analysis?
The gold standard when you have studies which have used different levels of the same explanatory variable is network meta-analysis (also known as multiple treatment comparison). If you do not want to delve into that level of complexity you have a number of more or less satisfactory ways of going forward. You could just select two levels and use them throughout while just ignoring all the other results. If you have all the raw frequencies you could, as you say, recompute unadjusted odds ratios for one comparison (say none versus more than none). If the unadjusted and adjusted ratios are close this might be convincing enough for your audience.
You should in general not use the unadjusted odds ratio from observational studies. If education were randomly allocated to subjects, an unadjusted (e.g. for key covariates) odds ratio would be only be slightly biased towards no effect and an unadjusted odds ratio may be okay. However, given that these studies presumably use observational data without random allocation to education levels, there are all sorts of selection biases etc. that typically make it impossible to attempt to say anything about the effect of education without some adjustments.
How to correctly use adjusted odds ratios is another question. A simple network meta-analysis treating them as totally different interventions is not quite right, because some of the education levels are subsets or finer splits of others. However, if we know the number of each people at each (finely split) level of education in each study, then presumably we could write down a model for how they would combine. I could imagine that someone has already developed and programmed (and published) such a model, but if so, I am not aware of it.