I have no doubt they can design and build them, given their successful ship and warship construction in the past.
It's not 1941 anymore! (And in any case the biggest issues plaguing Italian ships in 1941 was lack of modern radar and reliability of powerplants neither of which is an issue in 2021).
You realise the latest US frigate design (Constellation class) is based on FREMM which is a joint French-Italian ship.
Italian warships (and helicopters) are pretty good if not excellent. And their navy packs a fair bit of punch.
And unlike US or UK, Italy has maintained production of civilian ships and its ships are built to high standards (though as proven, deliberately driving them onto rocks to show off your captaining skills will still sink them).
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As for South Korean carrier, South Korea announced intentions to become a blue water navy as early as late 1990s though plans were already started to be formed in 1989 ( "Strategic Task Fleet" mentioned in the 1989 Joint Strategic Objectives Plan).
To help do this, South Korea has already:
1. Commenced exercises that saw longer deployment for its ships (started in 1990).
2. Purchased long range P-3C patrol/ASW planes starting in 1995.
3. Rapidly expanded submarine fleet since 1990s (Type 209 in 1993 followed by ultra modern Type 214 in early 2000s and new Dosan Ahn Changho class (3,750 ton) in 2022.
4. Commissioning of large air warfare destroyers (11,000 ton) and LHDs (19,000 tons) from mid 2000s.
5. Commissioning of large LST (2014)
6. Acquisition of longer range and increasingly larger logistics ships.
Current major fleet units are:
18 attack submarines (+9 additional on order)
12 destroyers (+3 additional on order)
15 frigates
11 corvettes
2 LHDs (helicopter carriers) –
4 large LSTs
16 x P-3C maritime patrol aircraft