It's complicated, but some of these extracts from British History Online might be of help.
The Charity Estates schools look to be possibilities, I expect Hall Green School the most likely. (The grammar school is too late and clearly 'Yardley' but was linked through the Charity). I've included the temporary board school too (this only took girls) as it might be of interest to others.
HALL GREEN YARDLEY CHARITY SCHOOL. In existence 1819, with 39 pupils.(2) Possibly identical with Hall Green School, q.v. In 1833 the master received £40 a year and a good schoolhouse etc. from the trustees of the Yardley Charity, and taught freely all children sent by them. He was allowed to take 4 paying boarders. A garden and playground were rented for the sch. by the trustees.(6) Received annual grant from 1881. A.a. 1881: 26.(1) Closed 1898 and pupils transferred to Hall Green Bd. Sch.(21) See also Yardley Charity Estates Sch.
HALL GREEN (TEMPORARY) BOARD SCHOOL. Hall Green Church Sch. There was a National Sch. at Hall Green in 1833, with 30 G in attendance.(3) The sch. was said in 1877 to have been built about 46 years before and established 40 years before (i.e. 1831 and 1837) and to have been thoroughly repaired 1877. It was rented and comprised one schoolroom. It was for G, I only. Fees 1d.–2d.(22) Received annual grant from 1881. Accom. 1881: 42. Transferred to Yardley Sch. Bd. 1890 as Hall Green (Temporary) Bd. Sch. Closed 1893 on opening of Hall Green Bd. Sch.(21)
YARDLEY CHARITY ESTATES SCHOOL, Yardley. Yardley Free Sch. A deed of Yardley Charity Estates of 1575 refers to the schoolhouse (see V.C.H. Worcs. iii. 238). In 1766 some of the income of the estates was set aside for the maintenance of a schoolmaster to teach the children of Yardley freely. He was to be unmarried and to have a rentfree house. In 1819 n.o.b. varied between 40 in summer and 70 in winter: the master was then said to refuse to take children who could not read or G unless they paid fees.(2) From then the sch. was for B only: there appears to have been no criticism of this.(6) In 1846 there was one schoolroom with n.o.b. 69.(30) By a Scheme of 1878 not more than £240 a year were to be spent on this sch. and the Hall Green Yardley Charity Sch. This was raised in 1882 to £280 and after the Hall Green Sch. closed 1898 it was all to be applied to this sch.(21) Received annual grants from 1879. Apparently enlarged 1894: (1) by 1907 there were 2 schoolrooms and 2 classrooms, one of each on each floor. Bd. of Ed. condemned building 1908, and sch. closed on opening of Church Road Council Sch.(21)
YARDLEY COUNTY GRAMMAR SCHOOL, Warwick Road, Tyseley. Yardley Council Secondary Sch. (sometimes known as Sparkhill Secondary Sch.) opened 1904 by Worcs. C.C. in Sparkhill Institute. New building in Warwick Rd. opened 1910 (V.C.H. Worcs. iv. 535). War damage repaired 1953.(28) It started as a mixed secondary sch. with a pupilteacher centre attached, although science and other facilities were considered inadequate for the latter, which closed 1913. Until 1910 partly maintained by Yardley Educational Foundation (see also Yardley Charity Estates Sch.). Under a Scheme of 1910 half Foundation's net income was devoted to maintenance of exhibitions at the sch. N.o.b. 1914: 180, 1961: 592.(28) In 1911 81 per cent. and in 1939 99 per cent. pupils came from elementary schs. Fees £3 until 1907, and £12 by 1939. In 1914 58 per cent. pupils paid fees, by 1939 none paid full fees. Average leaving age 1910–13 and 1938: 15–16 (ex inf. M. of Ed.).
References
2. Digest of Returns to the Select Committee on the Education of Poor, vol. ii, H.C. 224 (1819), ix (2).
3. Education Enquiry Abstract, vols. ii and iii, H.C. 62 (1835), xlii, xliii (relating to 1833).
6. Twenty-Sixth Report of the Commissioners for. . . Charities, H.C. 681 (1833), xix — King's Norton, Northfield, and Yardley parishes.
21. Ministry of Education School Files for Birmingham.
22. Ed. 7/128 — preliminary statements for Birmingham schools, arranged under the school numbers used in Board of Education lists.
28. Information provided by Birmingham Education Department. Accommodation figures for 1955 relate to January of that year, and those of 1961 to May.
30. National Society, Church Schools Enquiry, 1846–7. Copies of this and of the annual reports used were then in the possession of the society.