When कर्त्ता and कर्म are associated with कृत्-suffix then षष्ठी is implied (example?).
However, with अक and अ-ending कृत्-suffixes, षष्ठी is implied for both कर्त्ता and कर्म. For example, चिकीर्षा देवस्य कटस्य.
Some experts indicate षष्ठी for कर्त्ता and तृतीया/षष्ठी for कर्म - in the context of (non-अक and non-अ) कृदन्तs. For example, विचित्रा जगतः कृतिः हरेः / हरिणा.