6

I know 資格 is しかく, 活動 is かつどう and 許可 is きょか, but the 外 puzzles me here. Is it just がい?

2

As you suspect, しかくがいかつどうきょか.

An agreeing source (via pronunciation)

Edit: hah i'm too late. The same source is cited in the comments before me.

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