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VariantsToBinaryPed would seem to expect the fam file (first six cols of ped file) to describe the samples in the same order as the input VCF file: if they are not in the same order, it would appear to not correctly map sample IDs with the genotypes in the output binary PED.
I found this issue because I converted trio VCF files to binary PED, and then computed kinship coefficients using the binary PED file which showed that the relationships were wrong. If I fixed the .fam file so that the sample IDs were in the same order as the .vcf file and re-run the conversion to binary PED, then the kinship coefficients are as they should be given the pedigree.
This also made me wonder whether PhaseByTransmission has the same problem, but initial tests would seem to indicate that PhaseByTransmission may handle correctly the scenario where the sample order differs between the .fam file and the .vcf files.