The stored procedure returns the result set you want - why complicate it (and double the execution time)?
You may have simplified things a little to help - in which case, you could run the results of the stored procedure into a #temp table and join to it in your query:
CREATE TABLE #Projects (project_name VARCHAR(stringsize))
INSERT INTO #Projects (project_name)
EXEC STORED_PROC_NAME '%20%Catalog%'
SELECT g.project_name
FROM Groups g
INNER JOIN #Projects p
ON p.project_name = g.project_name
For fast, accurate and documented assistance in answering your questions, please read this article.
Understanding and using APPLY, (I) and (II) Paul White
Hidden RBAR: Triangular Joins / The "Numbers" or "Tally" Table: What it is and how it replaces a loop Jeff Moden