Yes, I had tried ENSEMBL Rest API. But, I could not get the correct information I was looking for using the API. The Entrezdirect option does work for extracting the taxon information. Sorry for not being very specific in my question earlier. By group in the question, I meant the NCBI-Blast name.
From this link i got the taxon-id for all the species. Then after entering the taxon-id here. I got the NCBI-blast name as output which I was looking for. However, doing this manually one by one would be tedious. @genomax can you please tell me if there is any way to extract NCBI-Blast names given the list of taxon-id.

To add further, I found out the grouping of species here. Is there any way I can find this information based on species name or ENSMBL IDs.