Learn English – Using Present Perfect and Perfect Gerund in the same sentence

Regarding finished past events, we use the

present perfect simple to relate events that occurred in a time period that has not yet finished

and the

simple past for events that took place in a time period that has ended.

American English generally allows for the use of either tense in these situations, while British English grammar insists on using the two tenses to distinguish them.

The arrest took place in the time period of "early this morning", which is apparently assumed to be over at the time of the utterance. So the relevant fact is not that the arrest is finished, but that the early morning is finished.

"This is the third time she's been arrested this year", on the other hand, occurs in an unfinished time period (if this year is not yet over), and so allows for (British: requires) the present perfect.

There's a lot of confusion around this, and a lot of half-true "rules" are taught.

The actual rule for the present perfect is fairly simple. The present perfect is a present tense, designating a present situation, so it cannot be used with a temporal expression which does not include the present, the moment at which the sentence is uttered.

Consequently:

^okThe directors have met is acceptable, because there is no temporal modifier. This merely asserts that the consequences of the fact of their meeting (whatever those consequences are) are now in play.

^okThe directors have met this week is acceptable, because the temporal modifier this week designates a timespan which includes the present: it runs from the beginning of the current week right up to the moment of speaking.

^∗ The directors have met last week is not acceptable, because the temporal expression last week designates a timespan which lies entirely in the past, ending with the beginning of the current week.

Note, however, that an entirely past temporal expression may be included in a present perfect sentence if it is 'bracketed' with commas or dashes or parentheses; it is then understood as a 'supplement', not integrated into the sentence but added to the sentence as an afterthought.

^okThe directors have met—just last week, as a matter of fact.

ADDED:
DamkerngT points out that you conclude that use of simple past with an indefinite temporal expression is incorrect. This is not the case. The simple past can be used with no temporal or with indefinite temporals:

^okThe directors met.
^okThe directors met sometime.
^okThe directors met before lunch.
^okThe directors met after that.

The only sort of temporal which cannot be used with the simple past is one which cannot be interpreted as lying entirely in the past, such as now or currently or tomorrow.