These letters are useless, and G is severely misused. I propose:

1) We abolish J.
1.1) A soft J sound (as in Je ne sais quois) shall be represented by Zh, following the pattern already established by S and Sh.
1.2) A hard J sound (as in Pajamas, Jump, etc.) shall be represented by Dzh. (A hard J sound is just a soft J sound preceded immediately by a D)

2) We restrict G.
2.1) Where it would normally represent a soft J sound (as in Peugeot) shall be replaced with Zh.
2.2) Where it would normally represent a hard J sound (as in Page, Stooge, etc.) shall be replaced with Dzh.
2.3) Where it would normally represent a G sound (as in Game, Ghost, Gimp, etc.) shall not be replaced.

3) We abolish C.
3.1) Where it represents an S sound shall be replaced with S.
3.2) Where it represents a K sound shall be replaced with K.
3.3) Where it is used in Ch to represent a Tsh sound, it shall be replaced with Tsh.
3.4) Where is it used in Ch to represent a [x] sound (as in Loch), it shall be replaced with Kh.

4) We abolish Q.
4.1) Where it represents a simple K sound (as in Qat, Quran, etc.) it'll be replaced with K.
4.2) Where it represents a Ky sound, it'll be replaced with Ky.
4.2.1) Where Qu is used to represent a Ky sound instead of just Q, Qu shall be replaced with Ky.
4.3) Where it represents a Kw sound, it'll be replaced with Kw.
4.3.1) Where Qu is used to represent a Kw sound instead of just Q (as in Queen, Squack, etc.), it'll be replaced with Kw.

5) We abolish X.
5.1) Where it is used to represent a Ks sound, it shall be replaced with Ks.
5.1.1) Except in American English, where it shall be replaced with Gz because of the difference in pronunciation.
5.2) Where it is used to represent a Z sound (as in Xenophobia), it shall be replaced with Z.

examples:

James will become Dzhames.
Cat and Qat will become Kat.
Queen will because Kween.
Peugeot will become Peuzheot, although I'd prefer Perzho.
Thrice will become Thrise.
Packet will become Paket.
Xylophone will become Zylophone.
Exit will become Eksit, unless you're in America, in which case it will become Egzit.
Chocolate will become Tshokolate.
Loch will become Lokh.

And so forth.

And I'm betting that you just read those examples out loud in your head, despite the pronunciation staying the same and only the spelling changing.