When you use scripts to handle your downloaded files, you want to be sure that the files downloaded by your Transmission daemon have the good permissions.
These files permissions are set in /etc/Transmission-daemon/settings.json in the variable umask.
Umask is normally represented in octal, but unfortunately, json does not support that, so we have to convert it to base 10.
The default value is 18 (in base 10) which correspond to the rights 022.
If you want your group to have all the rights you need to change it to 002 (2 in base 10).
So, now the code:
sudo /etc/init.d/transmission-daemon stop sudo nano /etc/transmission-daemon/settings.json
Edit the Umask value, Save and Exit.
sudo /etc/init.d/transmission-daemon start
Try with a new download if it works for you.