First, if you are working with a measurement system where you expect a constant CV over a range of necessarily positive values, then it's often best to work with logarithmically-scaled data instead of the raw values. The assumptions behind many statistical tests require that the magnitudes of residual values around mean values be independent of the magnitudes of the values themselves; roughly, the SDs should be independent of the mean values. That's not true with your data in their original scale, but if you expect constant CV among samples then it should be true on a log scale.
Second, the CV is the ratio of the SD to the Mean, and the sample SD is a biased estimator. So it would be inappropriate to average the necessarily biased CV values.
It's best to use variances rather than SDs for this type of analysis, as the variance of a sum of independent variables is the sum of their individual variances. Given your type of data, you could consider working with values transformed on a natural log scale, and then performing a simple analysis of variance (ANOVA) among the samples.
ANOVA will give a residual mean square, representing the estimated variance of observations around the corresponding sample mean values, pooled among all the samples. The square root of that residual mean square is an estimate of the SD around the mean values, and with natural-log-transformed data that square root directly provides a pooled estimate of the CV.
Note, however, that your estimate of the CV from just a few samples like this can be quite different from the true underlying CV. See this paper for the number of cases that need to be examined to obtain CV estimates with desired confidence limits.
