Which situation does NOT permit an extension of time to respond to an Office Action?

The scenario involving filing for a new continuation application does not permit an extension of time to respond to an Office Action because, under the rules of patent prosecution, extensions of time are generally associated with specific procedural actions directly related to the examination of an application.

When a continuation application is filed, it effectively creates a new filing that is not inherently tied to the original Office Action. The filing of new continuation applications is considered a new proceeding rather than a direct response to an Office Action. Therefore, extensions are not applicable in this context since continuation applications undergo their own examination independently from previous Office Actions received in earlier filings.

On the other hand, options like replying to a rejection from an Appeal, when an Examiner explains an amendment, and requesting an Oral Hearing are all activities that pertain to ongoing examination processes or procedural actions established under Patent Office rules. These situations allow for extensions because they involve direct interactions or responses to the Office's actions or requests that are subject to time constraints defined by the patent rules.