Is there a way to use an Index-Match formula where the Match needs to match on a substring in the cell?

For example, I have the following data
SOURCESHEET:
group position data
My Group: group1 1 this is my data
My Group: group1 2 more data
My Group: group2 2 la-di-da
My Group: group2 1 tiddly-winks
My Group: group3 1 Funky dance!
My Group: group3 2 More dance!
Another Group: group1 1 Data comes in
Another Group: group1 2 Data goes out

Of course, in another worksheet, I will want to match group1 and extract the value in the data column at that cell.
If I was performing an exact match, it would look like:
INDEX('SOURCESHEET'!A:C,MATCH('DESTSHEET'!A2,'SOURCESHEET'!A:A,0),3)
where 'DESTSHEET'!A2 has the value of "group1". Of course, this doesn't work because column the column I'm trying to match on has 'My Group: group1'.

Is there a formulaic solution to my situation?

As you may surmise, I will have to take that result and refine it against the Position column. But I'll should be able to figure that out after I solve this substring match issue.