Al-jawab billahi at-taufeeq (the answer with Allah's guidance)
Qunut, according to the definition of the Fuqaha’, “is the name of a Du’a (supplication) offered during prayer at a specific point while standing.” It is prescribed in Witr prayer before the Ruku), according to the opinion of the Hanafis.
It was narrated that Umar (RadhiyAllahu Anhu) prayed Qunut with the following words:
“Allaahumma inna nasta’eenuka wa nu’minu bika, wa natawakkalu ‘alayka wa nuthni ‘alayka al-khayr, wa laa nakfuruka. Allaahumma iyyaaka na’budu wa laka nusalli wa nasjudu, wa ilayka nas’aa wa nahfid. Narju rahmataka wa nakhsha ‘adhaabaka, inna ‘adhaabaka al-jadd bil kuffaari mulhiq..
(O Allaah, verily we seek Your help, we believe in You, we put our trust in You and we praise You and we are not ungrateful to You. O Allaah, You alone we worship and to You we pray and prostrate, for Your sake we strive. We hope for Your mercy and fear Your punishment, for Your punishment will certainly reach the disbelievers...)
(Al-Bayhaqi, 2/210, See also: Ibn Abi Shaibah, Chapter: The Qunut of Witr)
It is mentioned that Umar (RadhiyAllahu Anhu) used to recite this Du'a silently and sometimes aloud to teach the people. (Maraqil Falah Pg.143)
Regarding the issue of the Imam reciting Du'a Qunut (Allahumma Inna Nastainuka...) audibly or silently is one regarding which nothing has been narrated from the Imams of the Madhab. The Fuqaha attempted to ascertain what their views could be in this regard, by making Takhrij (extracting) of this Mas'alah from other related Masa'il and Qawaid (rules), however they were unable to come to any conclusive decision, as some felt that Imam Muhammad would feel it should be recited silently and Imam Abu Yusuf the opposite.
The early Fuqaha then looked at the students of Imam Muhamad and found that his students would recite it silently. This indicated that this was the view of Imam Muhammad. The early Fuqaha of the Madhab all gave fatwa on this view, which has been the Fatwa of the Madhab for the last 1000 years plus based on their understanding of the Usuls of the Imams of the Madhab and the practice of their students.
Thus, according to the most preferred opinion, Du'a Qunut should be recited silently and completely (i.e. the Imam, Muqtadi and Munfarid should all recite it silently).
(Raddul Muhtar, Chapter: Witr and Nawafil Vol.2 Pg.444, , Fataawa Hindiyyah, Chapter: Salatul Witr Vol.1 Pg.123)
Also, It was the scholars of Ma wara' al-Nahar ("what lies beyond the river," i.e., Oxus, or called Transoxiana which basically comprised modern Uzbekistan) who opted for Ikhfa (silent) with the Qunut of Witr.
Allah Ta'ala says,
“Call on your lord with humility and in private” (Al-A’raf, 55).
Prophet (Sallallahu Alaihi Wasallam) said, "The best Du'a is the Du'a in silence. (Ahmad Vol.1 Pg.176, Ibn Abi Shaibah Vol. 1 Pg.376)
(see: 'Ilaus Sunan, Chapter: Reciting the Du'a Qunut silently in Witr prayer Vol.4 Pg.1837-1839)
It is thus clear that the opinion of reciting it audibly isn't narrated from the Imams of the Madhab and neither was it the practice of the Mutaqaddimin (early Fuqaha).
With regards to Qunut Nazila (Allahummahdini Fiman Hadayta....) it is recited aloud.
Narrated Abu Huraira:
Whenever Allah's Apostle intended to invoke evil upon somebody or invoke good upon somebody, he used to invoke (Allah after bowing (in the prayer). Sometimes after saying, "Allah hears him who sends his praises to Him, all praise is for You, O our Lord," he would say, "O Allah. Save Al-Walid bin Al-Walid and Salama bin Hisham, and 'Aiyash bin Abu Rabi'a. O Allah! Inflict Your Severe Torture on Mudar (tribe) and strike them with (famine) years like the years of Joseph." The Prophet used to say in a loud voice, and he also used to say in some of his Fajr prayers, "O Allah! Curse so-and-so and so-and-so." naming some of the Arab tribes till Allah revealed:--"Not for you (O Muhammad) (but for Allah) is the decision." (3.128) (Bukhari, Chapter:Tafseer (Prophetic Commentary on the Qur'an))
Note: According to the Hanafis Qunut Nazilah is only recited in Fajr prayers at times of adversities and disaster. (See: 'Ilaus Sunan)
And Only Allah Ta'ala Knows Best.