I had to play with security groups lately for different clients using JIRA and from my understanding, the configuration of a project's templates (scheme, screens,etc..) requires a group tagged with [ADMIN]
The utmost group you can give a regular user is "Group (jira-administrators)" which has "Administer Projects" permission at the project level, which lets them add components,versions, and similar things but nothing close to what you want to achieve.
Maybe vote on this issue : https://jira.atlassian.com/browse/JRASERVER-3156
UPDATE July 2017 --------------------
Atlassian seems to have updated some of these settings with the release of jira 7.4.X for more info please read the full patch notes : https://confluence.atlassian.com/adminjira/jira-platform-releases/jira-7-4-x-platform-release-notes