I'm a biochemist and I usually compare the variability of my measurements in terms of coefficient of variation (CV) since I can "visualize" the deviations more easily in terms of percentages deviation from my mean value. Now, I measured an analyte 20 times in a sample in one month, and again 20 times the second month. Now I'd like to calculate the mean CV of those two sets of measurements. I know I can get a pooled sd and then maybe get the arithmetic mean of two means to calculate at the end mean or pooled CV, but I suppose I cannot do that, true? Can I just get mean CV (add, and divide by two?). Statisticians are usually in terms of squares, adding and square-rooting sd-s or variances, am I right? And for what is worth, I suppose also that if I combine all data in one set, I'd probably get different sd and CV?
I don't see any special value to averaging coefficients of variation (CVs).
Ideally your CV is consistent across datasets or groups or variables in which case you can underline that fact by citing a narrow range.
Conversely, if your CV is not at all consistent an average is not informative and how to get it a secondary issue. A wider issue is that the CV is a ratio, SD/mean, and ratios are often volatile and not easily or comfortably averaged. Otherwise put, the point you make that average CVs could be calculated in different ways shows up that the notion of average CV is not well defined.
In the great majority of cases where I have seen coefficient of variation used:
The implication is that you should be working on logarithmic scale.
Or it is not really helping.
Sample data that is real or realistic for you would allow a more specific answer.
In short, getting very different results is likely to be a sign that the data need closer analysis. Ignoring the difference by averaging is going in the wrong direction.