The key to this sentence is treating "management" consistently as one thing throughout, since a collective noun like this can be singular or plural depending on how the sentence uses it, but it has to stay the same choice all the way through.
Here, the sentence later uses "it," a singular pronoun, to refer back to "management." That choice means the first verb has to match: "has promised," not "have promised," since a mix of singular and plural forms for the same noun reads as inconsistent.
The second thing to check is the verb right after "will." An auxiliary like "will" is always followed by the base form of the verb, so it must be "will consider," never "will considered."
Checking every option against these two consistency points, the only version that keeps "management" singular throughout and uses "consider" in its base form is the one with "has promised... it will consider."
So, the correct answer is The management has promised that it will consider my appeal.
Saying these four sentences aloud, one right after another, makes it easy to hear where each one stumbles.
The stumble is different in each of the other three readings, but the fourth sentence reads cleanly all the way through.
So, the correct answer is The management has promised that it will consider my appeal.