In addition to what Gerry had already mentioned: of course, if the subsamples B and C were true random samples from the full sample, call it "population", meaning they are "representative" then the correlation-coefficients of the smaller samples are always estimators for that of the "population", and if you use two or more random subsamples the estimated population-coefficient is somehow an average.
But well, as you state your problem, it looks very likely to me that B and C are not such random-samples but are taken using some criterion. If such a criterion is existent then one should determine whether it distorts the randomness of the subsamples: if you take,for instance, B from the left edge of the whole data-cloud in a scatterplot and C from the right edge then the best-fit-lines in that subsamples may have completely different slopes and variances around them.
[update2] If such an averaging of correlations is actually meaningful in your problem (your subsamples are random and not too small) then I'd recommend to average the z-transforms of the correlation-coefficients. That means $$ r_{estr_{\text{est}} = \tanh(\frac{\sum_{k=1}^{s}\atanh(r_k)}{s}) tanh(\frac{\sum_{k=1}^{s}\tanh^{-1}(r_k)}{s}) $$ where $s$ is the number of samples, because that fisher-transformation approximates by conversion a correlation-coefficient into a z-variable (normal distributed, mean=0, infinite range) where the averaging over the arithmetical mean is more meaningful.
[update]
Here I show examples where the subsamples were taken randomly. I generated correlated data of a population with n= 2000, normal distributed with mean=0, stddev=1, correlation r~ 0.35 . I show the variation of the occuring correlations if random samples of n=20, n=50, n=100 are drawn. For each sample-size I did 500 experiments and documented the frequencies of occuring correlations r in steps of about 0.1.
sample-n: 20 avg r: 0.37760 experiments: 500
pop-n : 2000 pop r: 0.35247
low r high r freq
--------------------------------
-0.2023 -0.2023 1
-0.1807 -0.0948 8
-0.0878 0.0101 15
0.0205 0.1068 25
0.1112 0.2101 60
0.2123 0.3098 100
0.3113 0.4073 81
0.4109 0.5102 83
0.5109 0.6100 73
0.6107 0.7078 44
0.7122 0.7891 10
===================================
sample-n: 50 avg r: 0.36040
pop-n : 2000 pop r: 0.35247
low r high r freq
--------------------------------
-0.1011 -0.1011 1
0.0175 0.1027 9
0.1098 0.2022 55
0.2056 0.3027 108
0.3043 0.4027 150
0.4047 0.5030 124
0.5045 0.6024 45
0.6099 0.6982 8
===================================
sample-n: 100 avg r: 0.35657
pop-n : 2000 pop r: 0.35247
low r high r freq
----------------------------------
0.0504 0.0703 3
0.1139 0.2032 20
0.2054 0.3034 115
0.3055 0.4038 217
0.4047 0.4956 133
0.5046 0.5471 12
===================================
One can determine confidence-intervals for the correlations; that intervals narrow with increasing size of the samples.
But this all is only useful if the different samples are really random and not taken by some systematic criterion.
